FOUNDATIONS FOR YOUNG CHILDREN
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Indiana Academic Standards
Indiana Department of Education

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FOUNDATIONS FOR MATHEMATICS [TOP]

Preschool children are curious, independent, energetic, and eager to learn new things. This makes them excellent candidates for acquiring math concepts that will form a working foundation for more formal math learning in kindergarten and primary grades. Nowhere is it more true to say children learn by experience and discovery than in their acquiring math concepts. Adults have many opportunities to use naturally occurring events to stimulate curiosity and problem solving in order for children to begin to make the critical connection between living and learning. Adults also influence the child’s attitude and self-concept with regard to math processes.

Math concepts that are appropriate for preschoolers to begin working with are: numbers, volume, capacity, length, area, shape, space, time, and size. Much of the work will be discovering relationships through matching and comparing, filling and emptying, and measuring and manipulating. There are many opportunities (teachable moments) for adults to ask questions or make comments (e.g., “I wonder what would happen if…”). The most important learning in early years is the vocabulary that develops as a result of these adult-to-child and child-to-child interactions.


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FOUNDATION 1
NUMBER SENSE:
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Number Relationships

Children learn the meaning of numbers in the every day experiences the adult provides in the home, classroom, and nature.  The child needs opportunities to watch, play, and interact with adults and other children to learn number vocabulary and to discover number relationships.  Developing number sense means more than merely counting.  It involves the ability to think and work with numbers easily, to understand their uses, and describe their relationships.

YOU MAY SEE THE CHILD BEGIN TO:

Use the names for numbers.
Demonstrate number sense using number collections or sets of objects. 
Understand that numbers always represent the same quantity, regardless of the order or physical arrangement of the objects counted.
Use concepts of first and last.
Count series of objects in a group and tell the number (e.g., show me 5 beads). 
Have knowledge of quantity and some of the comparisons of quantity (e.g., all, some, none, fewer, more, less). 
Progress from inventive to accurate counting and recognizing and matching number symbols with the appropriate amounts. 
Use numbers to predict and make realistic guesses.

A CHILD CAN BE SUPPORTED BY AN ADULT WHO:

Scenario
Using names for numbers and counting

Mrs. Lee leads the children in playing a rhyming game, “Ants on a Log.”   Mrs. Lee says, “Three (or number based on ages of children) Little Ants Come Out to Play on a Sunny Day.”   The children dramatize being the ants and being the log with groups of children.  Because the children enjoy rhymes, they repeat the game with two ants and one ant.  Mrs. Lee labels a paper “Ants on a Log” for each child and asks the children to make a representation of what they played.  Mrs. Lee observes that Jimmy seems hesitant to use the crayons and paper.  She talks to him about what he wants to draw and suggests some ideas for getting started. 
 
For snack time, Mrs. Lee and the children make “Ants on a Log”.  Each child selects 1 stalk of celery, 2 spoons of peanut butter, and the number of raisins based on the age of the child.  Mrs. Lee demonstrates the activity and assists the children in selecting the food by modeling and counting with them.  The children make and eat their snacks.  Mrs. Lee and the children talk about healthy snack foods.

Social/Emotional:  Cognitive: Physical:  Self-help: Communication/Literacy:

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FOUNDATION 2
COMPUTATION:
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Counting, Sorting, Classifying, and Comparing Objects

Learning to model, explain, and use addition and subtraction concepts in problem solving situations begins with the opportunity for young children to count, sort, compare objects, and describe their thinking and observations in everyday situations. In building the foundation for computation, children need opportunities to observe adults and peers applying mathematical concepts and using problem-solving techniques. Including these concepts in their play and in adult-supported activities, enhances children’s understanding.

YOU MAY SEE THE CHILD BEGIN TO:

Experience one-to-one correspondence (e.g., 5 cups for 5 people at the table).
Understand the concepts of same, equal, one more, or less than.
Explore the use and meaning of currency and coins.
Attach meaning to visual and verbal uses of numbers (e.g., counts on fingers).
Count objects, sort, organize, and compare groups of objects.
Develop estimation skills related to quantity.
Model situations that involve the addition and subtraction of whole numbers, using objects, pictures, and symbols.
Seek help from peers or adults in solving a problem.

A CHILD CAN BE SUPPORTED BY AN ADULT WHO:

Scenario
Counting, sorting, and comparing objects

Mary operates an early care and education program in her home. Mary puts some colored blocks around the room in varying amounts (example: 4 red, 3 blue, 3 yellow, 2 green). She plays music while each child finds one block. When the music stops, the children gather in a circle. Mary calls out one color and asks all of the children who have that color to form a group. Billy goes to the wrong group, and Sean helps him find the right group. All of the children are grouped according to their colors. Mary asks children to count the number of children with blocks in their group. Then she asks, “Which group has more blocks, which group has less, and which groups have the same amount?” Billy thinks they are all the same. Mary has the groups line up one to one to show one more, one less, and the same.

Social/Emotional: Cognitive: Physical: Self-help: Communication/Literacy:

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FOUNDATION 3
ALGEBRA AND FUNCTIONS:

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Finding Patterns and Relationships

Young children build the foundation for finding patterns and their relationships by exploring environments that are rich in shapes, sizes, colors, and textures. They learn to identify and describe patterns using mathematical language when there are opportunities to sort, classify, and label things in their environment. Children need hands-on activities to explore and describe patterns and relationships involving numbers, shapes, data, and graphs in problem-solving situations.

YOU MAY SEE THE CHILD BEGIN TO:

Reproduce patterns of sounds and movements (e.g., clap, stomp, clap).
Recognize simple patterns of concrete objects (e.g., look at beads that are strung yellow, blue, yellow, blue, and identify the pattern).
Reproduce simple patterns of concrete objects (e.g., string beads that are yellow, blue, yellow, blue).
Predict what comes next when simple patterns are extended.
Explore attributes of objects and begin to sort by similar traits such as shape, color, size, or function.
Recognize objects arranged in a series and begin to place objects in order through trial and error.
Describe sequence of events and objects.
Recognize charts and graphs as a way of collecting, organizing, recording, and describing information.
Understand and explain what a graph shows.
Identify shapes (e.g., square, rectangle, triangle, circle, diamond).

A CHILD CAN BE SUPPORTED BY AN ADULT WHO:

Scenario
Recognizing and reproducing a simple pattern

Emma and her mother plant flowers in the spring. They have red and white petunias. Emma digs a little hole for a petunia and decides to plant the red petunia. She begins to pick up another red petunia, but her mother suggests the white goes next. After planting a white petunia, Emma sees the pattern and says, “Now we put a red one in. Red, white, red!”

Social/Emotional: Cognitive: Physical: Self-help: Communication/Literacy:

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FOUNDATION 4
GEOMETRY:

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Recognizing Common Geometric Shapes and Using Directional Words

In building the foundation for recognizing shapes and using directional words, children need opportunities to explore the size, shape, position, and movement of objects within their physical environment. Spatial reasoning (describing the position, direction, and distance of objects in relation to the child) begins as children become aware of their bodies and personal space within their physical environment. Children learn to recognize, draw, and describe shapes by manipulating, playing with, tracing, and making common shapes using real objects in a variety of activities.

YOU MAY SEE THE CHILD BEGIN TO:

See space and size relationships (e.g., putting puzzles together).
Recognize, describe, and name shapes (e.g., circles, triangles, rectangles, squares).
Use words that indicate where things are in space (e.g., beside, inside, behind, above, below, here, there, in, out, over, under, next to, near, far).
Explore geometric shapes.
Notice differences and begin to identify, describe, model, draw, and classify geometric shapes.
Recognize geometric shapes in the environment.
Build maps of surroundings using blocks, boxes, and other materials.

A CHILD CAN BE SUPPORTED BY AN ADULT WHO:

Scenario
Recognizing geometric shapes and using directional words

A theme about shapes emerged at ABC Preschool. The children used play dough to roll out circle, square, and triangle shapes. Mrs. Jackson placed three boxes around the room, each labeled with a different shape. The children put their matching shapes in the box. After all the children placed their shapes in the boxes, Mrs. Jackson asked the children to put the shapes in and out, over and under, beside and behind, or above and below the boxes. Grace did not understand the concepts of above and below as explained by Mrs. Jackson. Mrs. Jackson showed Sue a picture card of a child putting a hat on the shelf above the coat in the closet. She also demonstrated putting the triangle above the box with the other triangles and then below the box.

The children also played a game to find “hidden” circles, squares, and triangles that are turned different ways or have odd shapes. The children drew circles, squares, and triangles with their fingers. They traced over them with their fingers and drew the shapes on paper. During snack time, the children cut their peanut butter sandwiches into different shapes with circle, triangle, and square cookie cutters.

During circle time, the children identified items in the room shaped like circles, squares, and triangles. Jim has a visual impairment. In order to help him identify shapes, Mrs. Jackson had stencils of circles, squares, and triangles available for him. She used glue to outline geometric shapes for tracing and coloring.

Social/Emotional: Cognitive: Physical: Self-help: Communication/Literacy:

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FOUNDATION 5
MEASUREMENT:

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Time and Measurement Relationships

Children need many opportunities to explore and discover measurement and apply the results to real life situations in order to construct concepts of measurement. As children begin to use actual measurement instruments and explore measurement relationships, they develop a sense of measurement.

YOU MAY SEE THE CHILD BEGIN TO:

Estimate quantity, distance, weight, and length of familiar objects (e.g., temperature of room, weight of a gallon of milk).
Use familiar objects as measuring devices (e.g., finger width, arms length, foot length).
Use appropriate language to discuss activity (e.g., will use hot or cold when speaking about temperature related to weather, or heavy and light to describe weight).
Recognize time as a sequence of events that relate to daily life (e.g., in the morning, after snack).
Become aware of and use the conventional language of measurement (e.g., inch, mile, hour, degrees, cup, gallon).
Show an increasing awareness of conventional measurement tools and methods.
Realize some activities take longer than others.

A CHILD CAN BE SUPPORTED BY AN ADULT WHO:

Scenario
Making estimates and using measuring tools

It was a warm, sunny day outside. Mrs. Jones seized a teachable moment and took the children outside to talk about shadows. After they played and chased their shadows, Mrs. Jones called them together to talk about shadows and what causes them. The children discussed whose shadow was the longest shadow, shortest shadow, widest, etc. Carolina did not speak English very well, so Mrs. Jones made sure she used big gestures to demonstrate sizes of shadows. Mrs. Jones also helped her by pointing to objects, repeating phrases and names often. Mrs. Jones had taken a yardstick with her on the trip outside. She showed the children how to measure the shadows with this tool. She also showed them how to make their shadow look different.

On other occasions, Mrs. Jones can play the following games with the children that encourages the development of measurement concepts:

(1) “I am tall, I am small. Guess what I am now?”
(2) “If you have short hair, jump up and down. If you have long hair sit down.”
(3) “Children stand back to back and decide who is tallest/shortest, who has longest/shortest arms, and who has longest/shortest fingers.” Because Jimmy uses a wheel chair, Mrs. Jones only measures with body parts or ideas that enable him to participate fully along with the other children.

Social/Emotional: Cognitive: Physical: Self-help: Communication/Literacy:

Scenario
Building concepts of measurement, noting changes, and sequences

Mary read The Gingerbread Man to her son, Jimmy. After reading The Gingerbread Man, they make gingerbread cookies. Jimmy assists by measuring, stirring, rolling, cutting, decorating, and eating the cookies. Jimmy experienced some difficultly when rolling the dough. Mary helped him with hand-over-hand assistance so that he could roll out the dough and cut the cookies. Mary asks Jimmy to retell the story of The Gingerbread Man before bedtime.

Social/Emotional:
  • Interacts with adult and promotes self-esteem by doing grown-up things.
  • Takes turns and experiences cooperation.
  • Practices manners.
Cognitive:
  • Practices following directions.
  • Practices doing activities in sequential order.
  • Measures and notes changes.
  • Uses senses of smell, touch, taste, and sight.
Physical:
  • Performs motor skills including rolling, cutting, and decorating cookies.
  • Practices eye-hand coordination.
Self-help:
  • Learns good hygiene such as hand washing, cleaning utensils, and putting them away.
  • Assists with cleaning the floor.
Communication/Literacy:
  • Talks about the story and making cookies.
  • Provides opportunity to retell the story.

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FOUNDATION 6
PROBLEM SOLVING:

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Ability to Reason, Predict, and Problem Solve Through Exploration

When young children have experiences in collecting objects and information, as well as opportunities to organize, describe, and graphically represent these collections, they succeed in building a foundation for collecting and using data and thinking about issues of relationships in problem-solving situations. To build a foundation for solving problems, young children need opportunities to hear, use, and apply relevant vocabulary while formulating questions and possible solutions with others based on their observations and experiences.

YOU MAY SEE THE CHILD BEGIN TO:

Group interesting objects, name or describe groups found in the environment, and label or describe those collections.
Represent the data in a variety of ways.
Interpret information presented in graph form; draw and discuss information.
Develop and use systematic approaches to problem solving by testing new possibilities and finding solutions.
Look for and give clues.
Make predictions.
Describe similarities and differences between objects.
Explain how groups are made and describe thinking in how groups were made.
Make guesses related to quantity (e.g., “How many do you think you have?”).
Play with computational tools (e.g., rulers, measuring cups, calculators, abacuses, adding machines, or computers).
Explore concept of whole, parts, and parts that make a whole (e.g., cutting an apple in half and putting halves into whole-fractions).
Act out/draw/discuss data in a variety of ways.

A CHILD CAN BE SUPPORTED BY AN ADULT WHO:

Scenario
Resolving conflict through problem solving

Mrs. Jones’s class of 3 and 4 year olds were engaged in many activities around the room. Mrs. Jones circulated through the room observing and facilitating learning with the children. Sara was in the block corner building a house for her favorite doll. All of a sudden from the block corner came Sara’s loud voice, “Timmy stop that!” Timmy had knocked down Sara’s house. Mrs. Jones wandered over to the block corner to find Sara in tears. Timmy stood by looking at the damage that he had caused. While Sara’s tears were flowing she said to Timmy, “Why did you do that? I was building a house for my doll.” Timmy said, “I want those blocks to build a fire station for my big red truck.” Mrs. Jones facilitated conversation between Sara and Timmy on alternatives that Timmy could have used instead of knocking down the house. The three of them talked through this situation, and the two children came up with the solution of dividing the blocks between them in order that Sara could build her house and Timmy could also build his fire station. After the two had settled their differences, they sat down together and Sara built her house and Timmy a fire station for his fire truck. Note: Sara and Timmy came up with the solution to their problem. Mrs. Jones only facilitated their conversation by asking questions, not solving their problem.

Sometimes a child may need a physical prompt rather than verbal prompt to assist the child to enter discussion rather than acting out feelings with another child. For example, if Timmy continues to act out feelings rather than discuss a conflict, Mrs. Jones could establish a physical (concrete) reminder: Use of a foam ball could represent “use your own words” so when Mrs. Jones sees Timmy ready to use body language rather than words, she rolls the ball to Timmy as a reminder to use his own words. Foam balls are in the room so when Timmy is rolled one, the other children do not know Timmy is getting extra prompting.

Social/Emotional: Cognitive: Physical: Self-help: Communication/Literacy:

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RESOURCES AND REFERENCES 
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For Adults

Colorado Department of Education. (In Process). Building blocks to mathematics (DRAFT). Denver, CO: Colorado Department of Education.

Copeley, J.V. (2000). The young child and mathematics. Washington DC: NAEYC.

Copeley, J.V. (2000). Mathematics in the early years. Washington DC: NAEYC.

Fromboluti, C. and Rinck, N. (1999). Early childhood: Where learning begins mathematics. Jessup, MD: U.S. Department of Education, Office of Educational Research and Improvement, National Institute on Early Development and Education.

Janke, R.A. & Peterson, J.P. (1995). Peacemakers A, B, C’s for young children. St. Croix, MN: Growing Communities for Peace.

Levin, D.E. (1994). Teaching young children in violent times. Cambridge, MA: Educators for Social Responsibility New Society Publishers.

Ochlberg, B. (1996). Making it better. St. Paul, MN: Redleaf Press.

Rice, J.A. (1995). The kindness curriculum. St. Paul, MN: Redleaf Press.

Smith, C.A. (1993). The peaceful classroom. Beltsville, MA: Gryphon House.

For Children

Aylesworth, J. (1998). The Gingerbread Man. New York: Scholastic.

Carle, E. (1986). The Secret Birthday Message. New York: Harper Trophy.

Cohen, C.L. (1996). Where’s the Fly? New York: Greenwillow.

Crews, D. (1986). Ten Black Dots. New York: Mulberry.

Crimi, C. (1995). Outside, Inside. New York: Simon & Schuster.

Emberley, E. (1984). Ed Emberley’s Picture Pie 1: A Circle Drawing Book. Boston: Little, Brown.

Franco, B. (1997). Sorting All Sorts of Socks. Mountain View, CA: Creative Publications.

Guy, G. F. (1996). Fiesta! New York: Greenwillow Books.

Grossman, V. (1991). Ten Little Rabbits. San Francisco: Chronicle.

MacDonald, S. (1994). Sea Shapes. San Diego: Harcourt Brace.

Morris, A. (1995). Shoes, Shoes, Shoes. New York: Lothrop, Lee & Shepard.

Nagda, A., & Bickel, C. (2000). Tiger Math: Learning to Graph From a Baby Tiger. New York: Henry Holt.

Rotner, S., & Olivo, R. (1997). Close, Closer, Closest. New York: Simon & Schuster.

Schlein, M. (1996). More Than One. New York: Greenwillow.

Wallwork, A. (1993). No Dodos: A Counting Book of Endangered Animals. New York: Scholastic.

Wood, J. (1994). One Tortoise, Ten Wallabies. New York. Bradbury.

Websites

Department of Education, Office of Educational Research and Improvement, National Institute on Early Childhood Development and Education: http://www.ed.gov/teachers/how/early/list.jhtml

PBS Teacher Source math lesson plans and activities: http://www.pbs.org/teachersource/math.htm

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