Head Start Child Outcomes Framework
Domain 3: Mathematics

Domain Element: * Number and Operations

Indicators

• Demonstrates increasing interest and awareness of numbers and counting as a means for solving problems and determining quantity.
• Begins to associate number concepts, vocabulary, quantities, and written numerals in meaningful ways.
• Develops increasing ability to count in sequence to 10 and beyond.
• Begins to make use of one-to-one correspondence in counting objects and matching groups of objects.
• Begins to use language to compare numbers of objects with terms such as more, less, greater than, fewer, equal to.
• Develops increased abilities to combine, separate and name “how many” concrete objects.

Domain Element: Geometry and Spatial Sense

Indicators

• Begins to recognize, describe, compare, and name common shapes, their parts and attributes.
• Progresses in ability to put together and take apart shapes.
• Begins to be able to determine whether or not two shapes are the same size and shape.
• Shows growth in matching, sorting, putting in a series, and regrouping objects according to one or two attributes such as color, shape, or size.
Builds an increasing understanding of directionality, order, and positions of objects, and words such as up, down, over, under, top, bottom, inside, outside, in front, and behind.

Domain Element: Patterns and Measurement

Indicators

• Enhances abilities to recognize, duplicate, and extend simple patterns using a variety of materials.
• Shows increasing abilities to match, sort, put in a series, and regroup objects according to one or two attributes such as shape or size.
• Begins to make comparisons between several objects based on a single attribute.
• Shows progress in using standard and non-standard measures for length and area of objects.

* Legislatively mandated

DOMAIN 3: MATHEMATICS

     Mathematics helps children find meaning in their environment. As they learn to reason, connect ideas, and think logically, they gain important tools and concepts for making sense of the world. Mathematics relates to other curriculum areas, including science, social studies, art, and music. Last, but by no means least, math knowledge, interest, and skills are basic to children's success in school.

     Children develop math knowledge in part through their play and explorations of the world around them. Learning to recite the correct counting sequence or number facts (1 + 1 = 2) is possible for young children, but without concrete experiences they will not have a real understanding of what they are doing and why. In play, daily routines, and other meaningful activities, children question, analyze, and talk about their discoveries. When children see mathematics as part of everyday life, they find it useful, intriguing, and within their reach.

      Mathematics is an area where many English language learners accelerate because they can manipulate materials, as well as their bodies and hands, to practice math skills. If children know how to count in their home language they can easily transfer that knowledge of numbers into English.

Problem Solving and “Thinking Mathematically”

      To become mathematical thinkers, children need to learn mathematical concepts and relationships. Equally important, they need to learn basic but powerful aspects of problem solving and reasoning.(Because children use problem solving and reasoning in all knowledge areas, they appear in The Head Start Child Outcomes Framework under Approaches to Learning (Domain 7). They are discussed briefly here as well because they are integral to mathematics.) Children need to recognize, for instance, that there are many different ways to solve a problem and that more than one answer is possible.

      As children encounter problems in the classroom or at home, we can encourage them not only to tackle the problem but also to share their thinking with others. In Head Start, the goal is to create a learning environment in which children feel free to take risks and search for solutions to problems. Children become more conscious of their own reasoning and problem-solving strategies when teachers comment on what they are doing or ask about how and why they are doing it:

"Andre divided the playdough so that each person has the same amount. How did you do that, Andre?"

"I see a pattern in the chain you're making—red, blue, red, blue." "Now that you've run out of long blocks, what are you going to use to finish the last wall of your house?"

      Young children solve a lot of problems that arise in their everyday lives, relying on intuition or trial-and-error (NAEYC/NCTM 2002). On entering school, children will encounter a greater range of problems requiring careful thinking and systematic investigation. The skills and cognitive structures needed to solve problems in this deliberate, logical way are not well developed in preschoolers. However, in early childhood settings, teachers can work to enhance children's problem-solving dispositions and abilities (Copley 2000). In such learning environments, children become increasingly persistent, flexible, and proficient problem solvers—and they learn to enjoy solving problems.

      To promote children's mathematical thinking and learning, one of the most important efforts teachers and parents can make is to talk with them about problems, patterns, and mathematical connections and listen to what they say. Such dialogue helps children think about what they are doing and clarifies their thoughts (NCTM 2000). In addition, it improves children's math vocabulary, introducing them to words and phrases useful in mathematical reasoning and problem solving. Exploring ways of expressing mathematical ideas with words, diagrams, pictures, and symbols is also valuable for Head Start children.

     Head Start education staff play indispensable roles in promoting children's mathematical thinking and learning. They must have basic math knowledge and be alert to what children know and want to know about mathematics. Important roles for Head Start teaching teams include—

• creating learning experiences and environments to ensure that children "bump into interesting mathematics at every turn" (Greenes 1999, 46);
• investigating with children and observing what they do and say;
• answering children's questions and posing interesting questions and ideas for them to think about;
• providing many opportunities for children to represent problem solutions in a variety of ways; and
• serving as examples by modeling mathematics communication and investigation.

STRATEGIES
To promote problem solving

• Engage children in figuring out solutions to everyday situations. Talking about the problem, drawing children into the process of investigating and solving it, and asking how they came up with their solutions—all these strategies help to build eager, competent problem solvers. Make sure to give children time to think at each step along the way.
• Avoid solving problems for children, even though it often seems more practical. Whenever possible, adults should put the problem back in the hands of the children and give them time to resolve it. Of course, the teaching team continues to play the important role of modeling problem-solving steps and strategies and posing questions when children are stuck. But we need to let children do most of the thinking!
• Involve children in representing problems and quantities in ways other than talking. They can make diagrams or draw pictures to work through problems or represent solutions.
• Build into the curriculum lots of materials, situations, and activities where intriguing math problems are likely to arise.

     Finding ways to stir children's natural interest in math and problem solving is extremely valuable, but it is not enough. An effective early mathematics program is not a scattered assortment of unrelated activities—a pinch of this, a dash of that. Rather, a solid foundation in mathematics requires a planned, coherent curriculum that develops the core mathematical ideas and skills spelled out in the Child Outcomes Framework. These outcomes are based in part on the ideas and skills identified as important and achievable for young children in the National Council of Teachers of Mathematics report Principles and Standards for School Mathematics (NCTM 2000). These standards reflect a broad consensus among mathematics educators about the main ideas and content knowledge for young children to acquire.

Domain Element: Number & Operations
      Number sense involves the ability to think and work with numbers and to understand their uses and relationships. "Operation" is the formal mathematical term referring to addition, subtraction, multiplication, and division of numbers. (The 1998 Head Start reauthorization legislation refers to this Domain Element as numeracy.) Besides counting accurately and competently, children need to learn to see relationships between numbers and to take a specific number apart and put it back together. For example, there are several combinations of numbers adding up to five. As children acquire counting skills and become familiar with numbers, they are better able to understand other aspects of math.

      Experiences with estimation make quantity, numbers, and size more meaningful to children (NCTM 2000). As they move on in math, children will find estimation invaluable as a check on whether they have obtained a reasonable result in solving a problem. Young children are not able to make good estimates because they do not know enough about numbers and size, but they can begin to use this skill and will gradually improve.

STRATEGIES
To advance children's understanding of number and operations

• Encourage children to count all sorts of objects and events and to think about quantity and number. Teachers can use everyday experiences to promote concepts of number, counting, and one-to-one correspondence by posing questions such as, “Do we have enough chairs for everyone? How can we figure that out?” “Shall we count how many steps to the playground?” and “Who is third in line?” The questions listed [in Domain 4] can be used in mathematics as well as other Domains to stimulate children's thinking.
• Arrange materials and use games and verbal encouragement to involve children in—
• matching and sorting objects by color, shape, size, and other features;
• using one-to-one correspondence (one napkin at each person’s place at the table, for instance); and
• ordering a set of objects that vary in color, size, or another dimension (though not adept at seriation, preschoolers benefit from opportunities to try putting things in order).
• Draw children's attention to numbers around them and what they are used for, such as finding addresses, prices of objects, and shoe sizes.
• Use strategies to help children learn to count accurately and efficiently, such as—
• conveying to children that counting lets us know how many things there are in a group;
• pointing to each object in turn as the person counting (be it the child or the teacher) calls out each number name; and
• making use of fingers to count and encouraging children to do so.
• Highlight the relationships critical to developing number concepts and operations, such as the parts that make up a whole (a concept that underlies addition and subtraction). A teacher might say, "Brian is showing us how old his brother is by holding up five [fingers] and one. Can anyone think of another way to show six with their fingers?"
• Help children to become familiar with the skills and vocabulary of estimating, such as—
• using words regularly including more than, less/fewer than, about, near, approximately, and in between;
• asking children to estimate how much, how long, or how many ("How many shovelfuls do you think it will take to fill that bucket?"). During snack, sand or water play, art activities, and other opportune times, teachers can then encourage children to test for the actual answer; and
• making it a point to return to a problem type to allow children to try again. As children begin to make judgments closer and closer to the real count, they hone their estimation skills. Teachers should stress that it is not important for children to get the "right" answer, but to see how close they can come.

Domain Element: Geometry & Spatial Sense
     Geometry is the area of mathematics that involves shape, size, space, position, direction, and movement and describes and classifies the physical world in which we live. Young children can learn about angles, shapes, and solids by handling objects and looking around at the physical world. Spatial sense gives children an awareness of themselves in relation to the people and objects around them.

     Spatial sense and familiarity with shape, structure, and location enable children to understand not only their spatial world but also other mathematics topics (Clements, Sarama, & DiBiase 2002). For instance, the teacher can encourage a child to explore number concepts, such as even numbers, as he examines a cube and counts its faces (its sides) ("Someone told me he had a cube with 7 faces—do you think there could be such a thing?").

STRATEGIES
To build spatial sense and understanding of geometry

• Encourage children to identify different shapes (not just circle, square, and triangle but others as well) and three-dimensional figures as they draw, look at books, work with geometric puzzles, build structures in the block center, or take a neighborhood walk.
• Give children many opportunities to handle objects, such as blocks, boxes or containers, shape sorters, and puzzles.
• Let them climb in and out of boxes or large block structures; on or around outdoor equipment; and under, over, around, through, into, on top of, and out of different things to experience themselves in space.
• Encourage them to make new shapes by putting materials together and taking them apart in different arrangements. They can do this when cutting or folding paper, molding clay or building structures.
• Introduce spatial vocabulary, including—
• location and position words (such as on/off, over/under, in/out, above/below, in front of/in back of);
• movement words (such as up/down, forward/backward, toward/away from, straight/curved path); and
• distance words (such as near/far, close to/far from, shortest/longest).

Domain Element: Patterns & Measurement
     Taking note of patterns and relationships helps us understand the structure of things. In many areas, we find it useful and satisfying to anticipate what will come next. Patterns and relationships are found in science, music and dance, art, language arts (poetry, for example), and other areas. In mathematics, patterns are found in counting, basic number relationships, and in geometry. Understanding and identifying patterns and relationships means recognizing rhythm and repetition as well as sorting, categorizing, and ordering from shortest to longest, smallest to largest.

     Measurement is an important way for young children to look for relationships in the real world. We measure the length, height, and weight of an object using units like inches, feet, and pounds, and we measure time using hours, seconds, and minutes. In working with measurement, children focus on how big, little, long, or short things are and how to figure that out.

STRATEGIES
To promote children’s knowledge of patterns and measurement

• Increase awareness of patterns around the classroom and throughout the day. For instance, teachers—
• help children find patterns in designs and pictures, as well as in movement and in recurring events such as the daily classroom schedule, the days of the week, or the seasons of the year;
• engage children in creating and noticing patterns as they string beads; place shapes or blocks into arrays; and arrange other materials. Over time children can reproduce and create more complex patterns; and
• talk to children about patterns created or noticed in constructing with unit blocks, legos, and other construction materials.
• Let children figure out their own units for measurement (“Carlos is five cereal boxes tall”). Using standard units like inches, feet, and yards is only one way to measure. Children better understand these units when they have lots of experiences working with their own ways of comparing and measuring.
• Use many daily activities such as construction, cooking, woodworking, science, and other experiences that involve measurement. In any of these activities, teachers look for opportunities for children to make comparisons and measurements of volume, weight, length, and temperature.
• Provide simple experiences that help children begin to develop an understanding of time concepts, such as encouraging children to—
• compare one activity with another in terms of what takes more time. Start by asking simple questions like, "How long can you stand on one foot?";
• set simple time limits such as, "You can play for five more minutes and then we'll start clean-up". At first, children do not have a sense of how long a minute is, but gradually they get a better idea of time passing. An adult counting "one second, two seconds, three seconds..." or counting down from ten until a certain action or event is completed also helps children develop a sense of passing time; and
• start with ideas like "after lunch" or "after dinner" that provide solid milestones for children. Older preschoolers can begin to understand more abstract notions like yesterday, today, and tomorrow.
• Capitalize on children’s interest in comparing all sorts of things about themselves. In charts and graphs, they can see quantities displayed, which contributes to their understanding of numbers and their ability to make comparisons. You can create a chart recording children’s favorite desserts and pose questions such as, "Do more children like ice cream or cookies best?" Chart-making can be a very simple process. For example, placing one color form or sticker in a column to indicate each child’s preference creates a basic bar graph that makes sense to children.

     As adults share with children the kinds of experiences described here, they discover that mathematics is interesting and enjoyable to explore with preschoolers.