Praxis II Exam - Education of Young Child: Math Foundations

Problem Solving Skills
Being able to solve problems is fundamental to all other components of mathematics. Children learn the concept that a question can have more than one answer and a problem can have more than one solution by participating in problem solving activities. To solve problems, a child must be able to explore a problem, a situation, or a subject; think through the problem, situation, or subject; and use logical reasoning. These abilities are needed to not only solve routine/everyday problems, but also novel/unusual ones. Using problem solving skills not only helps children think mathematically, but also promotes their language development and their social skills when they work together. Children are naturally curious about how to solve everyday problems. Adults can take advantage of this inherent curiosity by discussing everyday challenges, asking children to propose ways to solve them, and asking them to explain how they arrived at their solutions. Adults can also invite children to propose problems and ask questions about them. This helps them learn to analyze different types of problems and realize that many problems have multiple possible solutions.

Common Steps That Prepare Children to Learn Math
The process of solving problems often involves the following steps: understanding the problem; coming up with a plan to solve the problem; putting that plan into action; and, finally, observing the outcome and reflecting on whether the solution was effective, and whether the answer arrived at makes sense. Solving problems not only involves learning this series of steps, but also requires children to develop the qualities needed to solve problems. Children who are able to solve problems have a number of characteristics. For example, children who are effective problem solvers are able to focus their attention on the problem and its individual component parts. They can formulate hypotheses about the problem/situation, and then test them for veracity. They are willing to take risks within reason. They are persistent if they do not solve a problem right away, and do not give up if their first attempt at solving a problem is unsuccessful. They maintain flexibility, and experiment with alternate methods. They also demonstrate self-regulation skills.

Using Problem Solving Skills in Daily Life
Young children continually explore their environments to unravel mysteries about how things work. For example, preschoolers use math concepts to understand that they have three toys, to comprehend that three fingers equals three toys, or to understand that two cookies plus one more equals three cookies. To do abstract mathematics in the future, young children will need two major skills that are also used to solve problems: being able to visualize a scenario, and being able to apply common sense thinking. Thinking and planning to achieve goals within the constraints of the properties of the surrounding environment is a natural behavior for young children. They will persist in their efforts to get an older sibling to stop another activity to play with them, to repair broken toys with tape or chewing gum, to manipulate a puzzle or plastic building blocks to get one uncooperative piece to fit, etc. The great 20th century mathematician and teacher George Polya stated that problem solving is “the most characteristically human activity.” He pointed out that problem solving is a skill learned by doing, and that developing this skill requires a great deal of practice.

Games/Activities That Encourage the Use of Problem Solving Skills
One method that has been found to enhance children’s reasoning skills is using adult-child conversations to play mental mathematics games. For example, once children are able to count beyond five, adults can give them basic oral story problems to solve (e.g., “If you have two plums and I give you two more, how many will you have?”). Using children’s favorite foods in story problems, which takes advantage of their ready ability to envision these foods, is a good place to start. Thereafter, adults can add story problems involving pets, toys, cars, shopping, and other familiar objects/animals/activities. Experts advise adults not to restrict the types of problems presented to a child based solely on the child’s grade level. Children can work with any situation if they can form mental imagery. Adults can sometimes insert harder tasks (e.g., problems involving larger numbers, problems involving division with remainders, or problems with negative number answers). Even toddlers can solve problems such as how to divide three cookies between two people. The division may not be fair, but it will likely be efficient. Adults should use the Socratic method, asking guiding questions to allow children to arrive at a solution to a problem themselves, rather than telling them a “right” answer.

Beneficial Practices of Playing Mental Math Games
Adults can use children’s favorite foods and toys to pose story problems to children that involve addition and subtraction. For example, they can ask them questions like “If I give you [this many] more, how many will you have?” or “If we take away [this many], how many are left?” It is better to ask children questions than to give them answers. It is important to use turn taking. In this method, the adult poses a story problem to the child, and then the child gets to pose one to the adult. Adults must try to solve the problem, even if the child makes up numbers like “bazillion” or “eleventy.” Games should be fun, not strictly factual like math tests. Adults can introduce age-appropriate story topics as children grow older. At the end of early childhood/around school age, children can handle the abstract algebraic concept of variables/unknown numbers (which some experts call “mystery numbers”) and use this concept in games. Adults can pose riddles where “x” or “n” is the unknown number, and children must use an operation (e.g., x + 4 = 7) to solve the riddle.

Reasoning Skills
Communicating with Children to Promote Mathematical Reasoning Skills
Adults reciprocally talk to and listen to children during communication that is focused on using mathematical skills like problem solving, reasoning, making connections, etc. To promote young children’s understanding, adults can express mathematical concepts using pictures, words, diagrams, and symbols. Encouraging children to talk with their peers and adults helps them clarify their own thoughts and think about what they are doing. Communicating with children about mathematical thinking problems also develops their vocabularies and promotes early literacy and reading skills. Adults should listen to what children want to say, and should have conversations with them. Communicating about math can also be accomplished through reading children’s books that incorporate numbers and/or repetition or rhyme. In addition to talking, adults can communicate math concepts to children by drawing pictures or diagrams and using concrete objects (e.g., blocks, crayons, pieces of paper, fingers, etc.) to represent numbers and/or solve problems. Children also share their learning of math concepts through words, charts, drawings, tallies, etc. Even toddlers hold up fingers to tell others how old they are.

Using Reasoning Skills to Understand and Apply Early Mathematical and Scientific Concepts
A major component of problem solving is reasoning. Children reason when they think through questions and find usable answers. They use reasoning skills to make sense of mathematical and scientific subject matter. Children use several abilities during the reasoning process. For example, they use logic to classify objects or concepts into groups. They follow logical sequences to arrive at conclusions that make sense. They use their analytical abilities to explain their own thought processes. They apply what they have learned about relationships and patterns to help them find solutions to problems. They also use reasoning to justify their mental processes and problem solutions. To support children’s reasoning, adults can ask children questions, give them time to think about their answers, and listen to their answers. This simple tactic helps children learn how to reason. Adults can also ask children why something is as it is—letting them think for themselves rather than looking for a particular answer—and listen to the ideas they produce.

Role of Representation Skills in Children’s Learning
Young children develop an understanding of symbolic representation—the idea that objects, written letters, words, and other symbols are used to represent other objects or concepts—at an early age. This is evident in their make-believe/pretend play, and in their ability to learn written language and connect it to spoken language. As children develop early math skills, representing their ideas and information they acquire helps them organize, document, and share these ideas and facts with others. Children may count on their fingers; create tallies using check marks/tick marks and/or words; draw pictures or maps; and, as they grow older, make graphs. Teachers must help children apply mathematical process skills as they use learning center materials. For example, when a child enjoys sorting rocks by color, the teacher can state that the child is classifying them, bridging informal math activities with math vocabulary. Asking the child how s/he is categorizing the rocks emphasizes math vocabulary. Asking the child after s/he finishes what other ways s/he could classify the rocks encourages problem solving.

Making Connections and Helping Children Transition from Intuitive to Formal Math Thinking
Children informally learn intuitive mathematical thinking through their everyday life experiences. They naturally apply mathematical concepts and reasoning to solve problems they face in their environment. However, one frequent problem among children when they begin formal education is that they can come to see academic mathematics as a collection of procedures and rules, instead of viewing it as a means of finding solutions to everyday, real-life problems. This view will interfere with children’s ability to apply the formal mathematics they learn to their lives in a practical and useful way. Teachers can help prevent this outcome by establishing the connection between children’s natural intuitive math and formal mathematics. They can do this by teaching math through the use of manipulative materials familiar to children. They can use mathematics vocabulary words when describing children’s activities, which enables children to develop an awareness of the natural mathematical operations they use in their daily lives. When a teacher introduces a new mathematical concept to children, s/he can give illustrative examples that draw upon the children’s actual life experiences.

Relationship of Mathematics to Everyday Life and Other Academic Subjects
We use math throughout our lives during everyday activities. There are countless examples and combinations of various mathematical concepts in the real world. Additionally, math concepts inform other academic content areas, including music, art, and the sciences. Therefore, it is important for children not to view math as an isolated set of procedures and skills. Children comprehend math more easily when they can make connections, which involve applying common mathematical rules to multiple, varied functions, processes, and real-life activities. For example, adults can ask children to consider problems they encounter daily and solve them. When a parent asks a child to help put away groceries, the child practices sorting categories of foods and packages, and experiments with comparative package sizes and shapes. Parents need not be concerned with what specific mathematical processes are involved, but should simply look for examples of math in everyday life and expose children to these examples on a regular basis. For example, pouring liquid into containers of various sizes and speculating which one will hold the most is an easy, fun activity that incorporates a number of skills and concepts, including estimation, measurement, spatial sense, and conservation of liquid volume.

Patterns and Relationships
Patterns are generally defined as things that recur or are repeated regularly. Patterns can be found in images, sounds, numbers, events, actions, movements, etc. Relationships are generally defined as connections or associations between things that are identified and/or described using logic or reasoning. Being aware of patterns and relationships among aspects of the environment helps us comprehend the fundamental structure of these aspects. This awareness enables us to predict what will occur next in a series of events, even before it actually happens. This gives us more confidence in our environment and in our ability to interact with it. We find patterns and relationships in such areas of life as art, music, and clothing. Math-specific activities like counting numbers and working with geometrical shapes, lines, arcs, and curves also involve patterns and relationships. When children understand patterns and relationships, they can understand repetition; rhythm; categorization; and how to order things from smallest to biggest, from shortest to longest, etc.

Adults can help young children develop their understanding of patterns and relationships in life by looking at pictures and designs with them, encouraging and guiding them to identify patterns within drawings, paintings, and abstract designs such as prints on fabrics and other decorative designs. When children participate in movement activities, including dancing to music, running, skipping, hopping, playing simple musical instruments, etc., adults can help them identify patterns in their own and others’ movements. Adults can encourage young children to participate in hands-on activities, such as stringing wood, plastic beads, or penne and other hollow dry pasta tubes onto pieces of string to make necklaces with simple patterns (e.g., blue-yellow-blue-yellow). As children grow older, adults can encourage them to create more complicated patterns. They can alternate a larger number of colors, and they can vary the numbers of each color in more complex ways (e.g., three blue, two yellow, one red, etc.).

Contribution of Number Sense and Number Operations to Math Comprehension
Counting is one of the earliest numeracy skills that young children develop. Even before they have learned the names of all the numbers, young children learn to count to three, then to five, etc.  However, number sense involves a great deal more than just counting. Number sense includes understanding the various applications of numbers. For instance, we use numbers as tools for conveying and manipulating information, as tools for describing quantities, and as tools for characterizing relationships between or among things. Children who have developed number sense are able to count with accuracy and competence. Given a specific number, they can count upwards from that number. They can also count backwards. They are able to break down a number and then reassemble it. They are able to recognize relationships between or among different numbers. When children can count, are familiar with numbers, and have good number sense, they can also add and subtract numbers. Being familiar with numbers and being able to count easily helps young children understand all other areas of mathematics.

Activities to Help Develop Number Sense and Numeracy Skills
As children complete their daily activities, it is beneficial for adults to count real things with children and encourage them to count as well. This helps children understand numbers by using their own experiences with objects in the environment, and gives them practice counting and using numbers. To help children understand that we use numbers to describe quantities and relationships, adults can ask children to sort objects by size, shape, or color similarity. They can also ask children to sort objects according to their differences (e.g., which object is bigger/smaller). Adults can also discuss with children how numbers are used to find street addresses and apartment numbers, and to keep score during games. To help children count upwards and downwards with efficiency and accuracy, adults can point out that counting allows us to determine how many items are in a group. Adults should point to each object as they count it. They can count on their fingers and encourage young children to do the same.  Adults should also help children count without repeating or skipping any numbers.

Counting
Counting is considered a math skill milestone for young children. Typical four-year-olds enjoy counting aloud. Experts identify three levels of counting. The first is counting from 1 to 12, which requires memorization. The second level is counting from 13 to 19, which requires not only memorization, but also an understanding of the more unusual rules for “teen” numbers. The third level is counting from 20 on. This process is very consistent, and the numbers are ordered according to regular rules. Experts in math education believe that at this level of counting, children are discovering a regular mathematical pattern for the first time, which is base ten (i.e. 20, 30, 40, 50, etc. are 2 tens, 3 tens, 4 tens, 5 tens, etc., and after the base a number between 1 and 9 is added). Researchers and educators in early childhood mathematics programs recommend encouraging children as young as four years old to learn to count up to 100. They find that doing this helps young children learn about and explore patterns in depth.

Perceiving and Identifying Shapes
The three levels of perceiving shapes that children typically move through sequentially are seeing, naming, and analyzing. Very young children recognize simple shapes like circles, squares, and triangles. As their cognitive and language skills develop, they learn the names for these shapes, and use these names to identify single shapes. The third level is analyzing each shape to understand its properties. Whereas identifying shapes visually is intuitive and based on association, analyzing their properties is more abstract, since a shape can have a number of different appearances. For example, three-year-olds can differentiate a triangle from other shapes. However, if you show them a very tall, skinny/short, wide/lopsided/crooked triangle, they will have trouble identifying it as a triangle. At the analysis level, children realize that a triangle has three sides, which are not necessarily equal in length. An activity that young children enjoy is closing their eyes, reaching into a bag of assorted shapes, finding a triangle by touch, and explaining why it is a triangle. This involves both the second and third levels of naming and analysis.

Spatial Sense and Geometry
Spatial sense is an individual’s awareness of one’s own body in space and in relation to the objects and other people around the individual. Spatial sense allows young children to navigate environmental spaces without colliding with objects and other people; to see and hear adequately, and to be aware of whether others can see and hear them; and to develop and observe a socially and culturally appropriate sense of their own and others’ personal space. Geometry is the area of mathematics involving space, sizes, shapes, positions, movements, and directions. Geometry gives descriptions and classifications of our physical environment. By observing commonplace objects and spaces in their physical world, young children can learn about solid objects and substances, shapes, and angles. Adults can help young children learn geometry by identifying various shapes, angles, and three-dimensional figures for them; asking them to name these shapes, angles, and figures when they encounter them in the future; and asking them to describe different shapes, draw them in the air with their fingers, trace drawings of the shapes with their fingers, and then draw the shapes themselves.

Activities to Help Develop Spatial Sense and Geometry
Because it involves many physical properties like shape, line, and angle, as well as abstract concepts, young children learn geometry most effectively via hands-on activities. Learning experiences that allow them to touch and manipulate concrete objects, such as boxes, containers, puzzles, blocks, and shape sorters, usually work best. Everyday activities can also help children learn geometry concepts. For example, adults can cut children’s sandwiches into various geometrical shapes and let children fit them together and/or rearrange them into new patterns. Children become better able to follow directions and navigate through space when they develop geometric knowledge and spatial sense. Adults can provide activities that promote the development of geometric knowledge and spatial sense. For example, they can let children get into and out of big appliance boxes; climb over furniture; and go into, on top of, out of, under, around, over, and through different objects and structures to allow children to experience the relationship between their bodies and space and solids. As they mature, children can play games in which they search for “hidden” shapes. Such shapes may be irregular, may lack flat bases, or may be turned in various directions.

Measurement
Measurement is the process of determining how long, wide, and tall something is physically and how much it weighs by using measuring units such as inches, feet, yards, square feet, ounces, and pounds. Measurement is also used to quantify time using units like seconds, minutes, hours, days, weeks, months, years, centuries, millennia, etc. Measurement is not just a formal means of quantifying size, area, and time. It is also an important method for young children to seek and identify relationships between and among things they encounter outside of school in everyday life. When young children practice measuring things, they are able to understand not only the sizes of objects and beings, but also their comparative sizes (i.e. how large or small something is compared to another object used as a reference). Furthermore, they are able to figure out how big or little something is on their own.

While it is obviously important for children to eventually learn standardized measurement units like inches, feet, yards, etc., adults can facilitate early development of measurement skills by letting children choose their own measurement units. For example, they might use their favorite toy to describe a playmate or sibling as “three teddy bears tall”; or they might describe a room as “seven toy cars long.” Similarly, when children are too young to know formal time measurements like minutes and hours, adults can support children’s ability to quantify time using favorite TV shows. For example, four-year-olds can often relate to the idea of one episode of a show (whether it is 30 minutes or 60 minutes long) as a time measurement. Adults can apply this with statements like, “Daddy will be home in one episode.” Numerous everyday activities, including grocery shopping, cooking, sewing, gardening, woodworking, and many others, involve measurement. Adults can ask children to help with these tasks, and then discuss measuring with children as they participate.

Measurement of Time
Younger children typically do not have an understanding of the abstract concept of time. However, adults can still help children understand that time elapses, and that we count/measure this process. For example, adults can ask younger children simple questions, such as “Who can stand on one foot longer?” This comparison strategy helps children figure out which of two or more actions/activities takes a longer/the longest period of time. Even when children do not yet understand what “five minutes” means, adults should still make such references (e.g., “You can play for five minutes longer, and then we must leave.”). Repeating such references will eventually help children understand that time passes. Adults can time various everyday activities/events and tell children how long they took. They can also count the second hand’s ticks on a watch/clock (e.g., “one second…two seconds…three seconds…”). This familiarizes children with counting, and with using counting to track the passage of time. Until children are old enough to understand abstractions like today/yesterday/tomorrow, adults can use concrete references like “after lunch” or “before bedtime.”

Fractions
Fractions are parts or pieces of a whole. While adults understand this and do not remember ever not understanding it, very young children think differently in this regard. As Piaget showed, children in the preoperational stage of cognitive development cannot perform logical or mathematical mental operations. They focus on one property of an object rather than all of its properties, a practice he called centration. Hence, if you cut an apple into pieces, very young children see that there are more pieces than there were before, and they believe that several apple pieces are more than one apple. They cannot yet comprehend the logical sequence of dividing an apple into fractions. To comprehend fractions, children must know what a whole unit consists of, how many pieces the unit is divided into, and whether the pieces are of equal size. Adults can help children understand fractions through informal sharing activities, such as slicing up a pizza or a pan of brownies, and/or equally dividing household/preschool chores and play materials.

Estimation
Estimation is making an educated or informed guess about a measurement when no actual measurement is available. As adults, we often make estimates about the sizes of objects when we do not know their exact measurements, about the amounts of substances we have not actually measured, and about the numbers of small objects in large collections when we have not actually counted the objects. However, young children are in the process of learning the concepts of sizes and numbers. Children must comprehend concepts of comparison and relativity (e.g., larger, smaller, more, less, etc.) before they will be able to make accurate estimates. When children start to develop the ability to estimate amounts or sizes, this process helps them learn related math vocabulary words, such as “about” or “around,” and “more than” and “less than” [something else]. Through estimating, they also learn how to make appropriate predictions and arrive at realistic answers. It is important for young children to learn how to make estimates, to recognize when it is appropriate to apply the estimation method, and to recognize when their estimates are reasonable.

Activities to Help Develop Estimation
To accustom young children to the idea of estimating, adults should regularly use words related to estimation in their conversations with children (e.g., “around,” “about,” “approximately,” “near,” “more than [some other amount or number],” “less than [some other amount or number],” “between [two numbers or amounts],” etc.). During everyday activities like shopping or eating, adults can ask children to estimate amounts of foods, numbers of items, or lengths of time. Later, adults can help children compare the actual outcome with their original estimate. This process helps children learn to make realistic/reasonable estimates. Activities promoting estimation skills can be very simple. Adults can ask children, for example, to guess which of their friends is tallest, and then test the accuracy of the guess using real measurements. When children grow older, adults can write down estimates and real measurements, and can then repeat the exercise described above or present a similar one. With repetition, children will eventually begin making more accurate estimates. The goal is not for children to come up with exact measurements, but ones that are close to actual amounts/numbers. Giving children opportunities to practice improves their estimating skills.

Probabilities and Statistics
In general, when people work with statistics, they present them in graphs or charts to organize them, interpret them, and make it easier to see relationships among individual statistics. Graphs are a visual alternative that depict mathematical information and show relationships among individual statistics, especially changes over time. Graphs also allow for the comparison of different groups. Probabilities indicate the likelihood that something will happen. Adults use probabilities to predict things, such as people’s risks of developing or dying from various diseases or medical conditions; the chances of accidents; children’s risks of experiencing academic difficulties, dropping out, or developing emotional and behavioral disorders; and the chances that a certain area will receive rain or snow. Scientists use probabilities to predict the likelihood of various behaviors or outcomes they are studying. They use statistics to show the numbers and proportions of responses or results obtained in research studies. Calendars are one type of chart. Adults can help children use them to organize daily and weekly activities, and to understand how we organize information.

Charts and Graphs
According to experts, almost every daily activity can be charted in some way. For example, adults can help children peel the little stickers off of plums, bananas, etc. and stick them to a piece of paper/poster board divided into columns. After a week, they can count each column to determine how many pieces of each kind of fruit they ate. Similarly, adults can show children how to use removable stickers or color forms to document the number of times they performed any daily activity. For example, children could place a color form near the telephone every time it rings and/or every time somebody picks it up to make a call. They could also place a color form near the front door every time somebody comes in, goes out, and/or rings the doorbell/knocks. This enables children to count the number of times given events occur by recording them. Some children are better able to understand math by viewing and making graphs. This is because creating graphs involves representing quantities visually instead of just listing numbers.

Rational Numbers and Irrational Numbers
In mathematics, rational numbers are numbers that can be written as ratios or fractions. In other words, a rational number can be expressed as a fraction that has a whole number as the numerator (the number on top) and the denominator (the number on the bottom). Therefore, all whole numbers are automatically rational numbers, because all whole numbers can be written as fractions with a denominator of 1 (e.g., 5 = 5/1, 68 = 68/1, 237 = 237/1, etc.). Even very large, unwieldy fractions (e.g., 9,731,245/42,754,021) are rational numbers, because they can be written as fractions. Irrational numbers can be written as decimal numbers, but not as fractions, because the numbers to the right of the decimal point that are less than 1 continue indefinitely without repeating.  For example, the value of pi (π) begins as 3.141592…, and continues without end. The square root of 2 (√2) = 1.414213…. There are an infinite number of irrational numbers between 0 and 1. However, irrational numbers are not used as commonly in everyday life as rational numbers.

Cardinal, Ordinal, Nominal, and Real Numbers
Cardinal numbers are numbers that indicate quantity. For example, when we say “seven buttons” or “three kittens,” we are using cardinal numbers. Ordinal numbers are numbers that indicate the order of items within a group or a set. For example, when we say “first, second, third, fourth, fifth, etc.,” we are using ordinal numbers. Nominal numbers are numbers that name things. For example, we use area code numbers along with telephone numbers to identify geographical calling areas, and we use zip code numbers to identify geographical mailing areas. Nominal numbers, therefore, identify categories or serve as labels for things. However, they are not related to the actual mathematical values of numbers, and do not indicate numerical quantities or operations. Real numbers include all rational and irrational numbers. Rational numbers can always be written as fractions that have both numerators and denominators that are whole numbers. Irrational numbers cannot, as they contain non-repeating decimal digits. Real numbers may or may not be cardinal numbers.

Integrating Math into Everyday Activities and Using Early Childhood Math Curricula
Integrating math into the context of everyday activities has been the philosophy of early childhood math education until recently. For example, when teachers have children line up, they ask them who is first, second, third, etc. to practice counting. When children play with blocks, teachers ask them to identify their shapes and whether one block is larger/smaller than another. During snack times, teachers help children learn 1:1 correspondence by having them place one snack on each plate. These activities are quite valuable. However, some educators maintain that they are insufficient when used on their own, because in larger classes it is not always possible to take advantage of “teachable moments” with every child. Therefore, this educational approach cannot be applied systematically. These educators recommend that in addition to integration strategies, EC teachers should use a curriculum. The HighScope curriculum, the Creative Curriculum, and Big Math for Little Kids are just a few examples.  Many teachers combine several curricula, selecting parts of different programs. Using a curriculum allows teachers to use a more planned approach to integrate math into all activities.

Clinical Interview
Background, Method, and Advantages
Clinical interviews have long been used by individual and family therapists, as well as by researchers. Piaget used them along with observations and case histories to understand young children’s thinking as he formulated his cognitive developmental theory. Interviewers ask structured/semi-structured/open-ended questions and listen to the responses, often recording them for accuracy. This method gives the interviewer a way to find out what the respondent is thinking and feeling inside, which cannot be determined by observing outward behaviors alone. In educational settings, a teacher might ask a child questions like, “How did you do this?” “What is happening now?” “Can you tell me more about this?” “Why are you doing this?” “What are you thinking about now?” etc. Flexible questioning helps uncover the child’s thought process, which is what is leading him/her to engage in specific behaviors. Just observing the behaviors alone does not allow the child to express his/her knowledge. While fully interviewing each child in a classroom is not practical, teachers can adapt this method by asking clinical interview-type questions as part of their instruction.

Using Questioning
Teachers can gain a lot of information and insight about how children are learning math concepts by observing their behaviors. For children to actually express their knowledge and thinking processes, however, teachers must ask them questions. For example, when a teacher introduces new shapes to young children, s/he can ask them the shapes’ names, how they differ from one another, and why they think the shapes differ. Teachers can then use children’s various responses to elicit further responses from them. This technique requires children to use language in significant ways during math activities. Therefore, these activities not only teach math skills, but also promote literacy development. Asking clinical interview-type questions promotes children’s development of math communication skills, one of the essential components of math education. Additionally, being able to put one’s knowledge and thoughts into words is a skill that is very important in all areas of education, not just math education. Using clinical interview-type questions helps children learn to use language to explain their thinking, share ideas, and express themselves, promoting and strengthening children’s awareness of the functions of mathematical language.

Characteristics of Young Children’s Thinking and Learning That Inform EC Math Curricula
Young children think in concrete ways and cannot understand abstract concepts, so effective EC math curricula typically use many concrete objects that children can see, feel, and manipulate to help them understand math concepts. Young children also naturally learn through exploring their environments, so good EC math curricula have many exploration and discovery activities that allow and encourage hands-on learning. In everyday life, young children start to observe relationships as they explore their surroundings. They match like objects, sort unlike objects, categorize objects, and arrange objects in simple patterns based on shared or contrasting properties. They start to understand words and phrases like “a little,” “a lot,” “more,” “less,” and “the same [as…].” Preschoolers use available materials such as sticks, pieces of string, their feet, their hands, their fingers, etc. as tools to measure objects. They also use rulers, measuring cups, and other conventional tools. They use their measurements to develop descriptions, sequences, and arrangements, and to compare various objects.

Activities That Help Children Develop Spatial Awareness
When preschool children build structures with blocks and put together pieces of puzzles during play, they are not only having fun, but are also developing spatial awareness. The relationships of objects to each other and within space are important concepts for children to learn, and serve as a foundation for the principles of geometry and physics that children will learn later. When they are moving around, preschoolers begin to notice how other people and objects are positioned in space, and how their own bodies move through space in relationship to objects and other people. This type of spatial awareness supports children’s developing gross motor skills, coordination, and social skills. Young children can and should learn a number of math concepts and skills, such as the ones recommended by preschool math curricula like the High Scope program’s “Numbers Plus” preschool mathematics curriculum. These concepts and skills include number symbols and names, counting, shapes, spatial awareness, relationships of parts to the whole, measurement, units, patterns, and analyzing data.

Activities and Games That Make Learning Fun
 

Button Board
By gluing buttons of various sizes and colors to a piece of cardboard, teachers can initiate a number of activities that help preschoolers learn math concepts while having fun. Preschoolers are commonly learning shapes and how to draw them. Teachers can give children lengths of string/twine/yarn or long shoelaces and show them how to wrap them around different buttons to form shapes like rectangles, triangles, and squares. To practice counting and 1:1 correspondence, teachers can ask children to wrap their string around a given number of buttons. Preschoolers need to learn the concept that spoken number words like “five” can equate to a group of five concrete objects (such as buttons), and this activity promotes that learning. The button board is also useful for giving preschool children practice with sorting or classifying objects into groups based on a common characteristic. For example, the teacher can ask children to wrap their pieces of string around all the big buttons, all the little buttons, only the red buttons, only the blue buttons, etc.

Beanbags and Hopscotch
Teachers can encourage preschool children’s counting and number development by creating a grid on the floor with the numbers 1 to 10 using masking tape, construction paper, and markers. Teachers could also draw the grid outdoors by drawing on pavement with chalk. The teacher arranges the numbers in ascending order within the grid of 10 squares/rectangles. S/he asks the children if they can name these numbers. The teacher provides beanbags. Each child gets a chance to throw a beanbag into any one of the numbered squares. Children can see how far they can throw and/or practice their aim. Each child names the number inside the square/rectangle where his/her beanbag lands. The children then play a version of hopscotch by hopping from numbered square to square, collecting their beanbags, and then hopping back. If desired, the teacher can write the number each child’s beanbag lands on onto a “scoreboard” graph. Children will observe his/her writing the same numbers found on the floor/ground onto a “scoreboard.” Teachers can review learning after the game to assess whether children can count using number words, name selected numbers, and throw accurately with consistency.

Reusing Sectioned Plastic Trays
A teacher can wash and reuse the compartmentalized plastic trays from the grocery store that are used for vegetable and fruit to create a preschool counting activity. The teacher supplies beads, pennies, erasers, or other small objects, as well as about a dozen sticky notes. S/he writes a number on each note. For older preschoolers, the teacher can write the numeral and the word (e.g., “7” and “seven”). For younger children, the teacher can write the numeric symbol (“7,” for example) plus seven dots or other marks as a clue to that number symbol. The teacher puts one numbered note in each compartment and the supply of small objects in the central dip compartment. Then, s/he guides each child to transfer the correct number of each small object to the correct compartment. The child should count aloud while transferring each small object, and should repeat this process until all compartments with a numbered sticky note have the correct number of objects. Children can then repeat the process to practice and perfect their counting, or the teacher can place notes with different numbers in the tray’s compartments.

Fishing for Numbers
Teachers can help preschoolers practice identifying numbers and counting by creating a fun “fishing for numbers” game. Teachers cut 10 fish shapes that are about 6 inches long from pieces of construction paper that are different colors. Teachers then write a single number between 1 and 10 on each “fish.” Near each fish “mouth,” the teacher punches a hole and inserts a paper clip through it. The teacher makes “fishing rods” by tying strings to dowels and gluing a magnet to each string. After spreading out the fish so the children can easily see the numbers, the teacher assigns each child a number and they “fish” for it, picking up the fish by bringing the magnet close to the paper clip. The children then “reel in” their catches. This gives children practice correctly identifying number names. The game can be adapted for more advanced math concepts as well. For example, the teacher can cut out fish shapes of various sizes and have children “fish” for larger/smaller fish. The activity can also be adapted to promote literacy development. The teacher can write letters instead of numbers on the fish to give students practice with alphabet recognition, or s/he can write a Dolch word/sight word on each fish to give students practice recognizing and identifying important vocabulary words.

Collages
Fundamental math skills that prepare preschoolers for kindergarten include shape recognition. To introduce children to an activity they will view as fun rather than as work, teachers can show children how to make a collage of a familiar figure. This will also give children the opportunity to experiment with an artistic process. For example, they can create a Santa Claus or an Easter Bunny as a holiday art project. They can make collages of other imaginary/real people for various events/seasons/topics.  Teachers cut out paper templates, including circles for heads, triangles for hats, squares for bodies, and narrow rectangular strips for limbs. First, they help children name each shape. They have each child trace the template shapes onto paper and cut them out with child-safe scissors. The teacher then instructs children to arrange their cutout shapes on a piece of cardboard/construction paper. Once they are in the correct positions, the children glue the shapes in place. Teachers can subsequently teach additional shapes (octagons, ovals, etc.), challenging children to make new, different collages.

Grab Bag
Young children learn to name numbers in a way that is similar to how they learn to recite alphabet letters. However, learning to associate number symbols with concrete objects in the real world environment is a major advance in their cognitive development. The concept of 1:1 correspondence entails matching number symbols to the quantities they represent, an essential early math skill. Teachers can support the development of this math skill with a simple “grab bag” game youngsters enjoy. The teacher writes a number from 1 to 10 on each of ten cards, folding each card in half and putting them into a paper lunch bag. The teacher provides each child with a handful of pennies/play coins/buttons/little blocks to use as counting tokens. Each child takes a turn closing his/her eyes and pulling a card out of the bag. The child reads the number on the card, counts out the corresponding number of pennies/tokens, and puts them with the card. As children learn, teachers can place additional and/or different numbers (e.g., 11 to 20) in the grab bag. To promote the development of early literacy skills, teachers can also include the name of the number on each card.

Pattern Resist Art
A significant mark of progress in early math skills development is the ability to not only identify various shapes, but also to draw them. Once young children develop this ability, they typically want to practice it all the time. Teachers can encourage this by helping children make pattern resist paintings. The teacher tapes white paper to children’s tables/trays, gives them crayons, and invites them to fill the paper with drawings of different shapes of various sizes and colors. Teachers can introduce young children to new shapes (e.g., ovals, stars, crescent moons, etc.) by drawing them on separate pieces of paper for children to look at and copy. Then, the teacher replaces the crayons with water, watercolor paints, and brushes; shows the children how to dip brushes into paint and water to dilute the colors; and allows them to paint over their crayoned shapes, covering all the white paper with color. The children see the shapes show through the paint, creating the pattern resist. Dipping brushes and diluting various colors also develop children’s color recognition skills and their hand-eye coordination.

Ice Cube Necklaces
In hot weather, making ice cube necklaces is a fun activity that helps young children cool off while learning to sequence objects. The activity also helps children develop their manual motor skills and learn about liquid and solid states of matter. Regular ice cube trays are fine; those with “fun-shaped” compartments are even better. The teacher cuts plastic drinking straws so that they will fit into each ice cube compartment. The children participate, watching and/or helping pour water into trays and adding various food colorings/fruit juices. The teacher places one straw clipping into each compartment. While putting the trays into the freezer, the teacher tells children that 32° Fahrenheit/0° Celsius is the temperature at which water freezes. Children practice making scientific observations by noting how long the water takes to freeze. They empty the cubes into a big bowl. The children put on bathing suits or other clothing that can get wet, and the class goes outdoors. The teacher provides strings that are knotted at one end, and calls out a color pattern (e.g., one blue cube, then a yellow cube, etc.). Children follow the teacher’s instructions to create color-patterned necklaces they can tie, wear, and watch melt.

Red Rover
Red Rover is a good game for groups of children who are attending parties or playing outdoors at parks/playgrounds. Two teams take turns calling and roving. The child called runs to the other team and tries to fit into its line. If successful, s/he gets to call another player to bring back to his/her home team. If not, s/he joins the opposite team. The game continues until one team has no more members. Teachers can adapt this game to teach shape recognition by cutting out various shapes from construction paper of different colors and pinning a shape to each child’s shirt. In large groups, more than one child can have the same shape or color. Instead of children’s names, the teacher instructs players to use shapes and colors when calling (e.g., “Red Rover, Red Rover, blue circles come over!”). This supports the development of shape and color recognition skills. Teachers can vary action verbs (e.g., “….hop over/jump over/skip over”) to support vocabulary development and comprehensive skills. When children perform such movements, they are also practicing and developing gross motor skills.

Counting on Fingers
A common practice among preschool children is counting on their fingers. Young children learn concretely before they develop abstract thought, so they must have concrete objects to work with to understand abstract mathematical concepts. They use their fingers to count because fingers are concrete. A simple activity that allows children to continue finger counting while removing additional visual support is “blind finger counting.” Using eyesight to count objects we can see is relatively easy. However, when children cannot see objects, they must learn to count mentally instead. This allows them to take another step in their progress from concrete to abstract thinking. To count mentally without visual reinforcement takes practice. Teachers can tape a shoebox lid to the box and cut a small hole in it. Children can fit a hand through the hole, but cannot see inside. Children close their eyes; the teacher drops several small objects into the box; and each child reaches in, counting the objects using only touch. Varying objects and quantities maintains the fun of this activity.

Sorting and Categorization
One of the major learning accomplishments of young children is being able to identify similarities and differences among objects. Developing this ability enables children to sort like objects into groups, and to place objects into categories based on their differences. When preschoolers compare and contrast objects, they demonstrate an important early step in the development of critical thinking, analytical, and problem solving skills. For an easy, entertaining guessing game, adults can select assorted household items familiar to children and put them into a bag/pillowcase. They then give children various clues (e.g., “I stir lemonade with this…,” “It’s made of wood,” “We keep it in the kitchen drawer…,” etc.) and ask them to guess which items are in the bag. It is important to give young children one to two minutes to consider each clue before they make a guess. Adults repeat clues when children guess incorrectly. If children guess correctly, they are allowed to look inside the bag. Youngsters greatly enjoy seeing that the object they guessed is actually inside the bag. Adults can gradually make the game more challenging by beginning with very common objects, and then eventually progressing to more unusual ones.

Baking Cookies
Young children are typically curious about adult activities like baking. They usually want to know more about the process, and often ask many questions. They also love to be included and to participate, frequently offering/asking to help. Letting them help builds their self-esteem and self-efficacy (i.e. their confidence in their competence to accomplish a task). Adults can allow children to help while also providing instruction and practice with shape recognition, measurement, sorting, and categorization.  The adult prepares a favorite cookie recipe. Some children can help measure ingredients, which helps develop the math skill of measurement. With the dough rolled out, children use cookie cutters of various shapes. Recognizing, naming, and selecting the shapes promote the development of shape recognition skills. Adults “shuffle”/mix the baked cookie shapes and have children separate cookies with like shapes into groups, which promotes sorting skills. Having children identify similar/different shapes, sizes, and colors promotes categorization skills. Arranging cookie shapes into patterns for children to identify promotes pattern recognition skills, which are necessary to the development of math skills and many other skills. Giving each child a cookie to eat afterward is naturally reinforcing.

Creative Crafts
Prerequisite abilities that young children need in order to develop early math skills include the ability to identify, copy, expand, and create patterns; as well as the ability to count. Adults can promote the development of these skills by giving children a craft project and introducing them to an interactive game they can play using their crafts. First, the children paint six ping pong balls red on one side to make red-and-white balls. Then, the children paint six ping pong balls blue on one side to make blue-and-white balls. Once the paint dries, the adult puts several balls into an egg carton so that one color is face up. The adult starts making a simple pattern (e.g., two white, then two red, then two blue), and asks each child to continue the pattern. Then, the adult allows each child to create his or her own original color patterns. Once a child masters creating patterns using solid colors, he or she can then use both the white and colored sides of the balls to create more complex patterns. Children can design an infinite number of patterns, which are often quite artistic.

Shape Matching Games
In one type of shape matching game, EC teachers help children make a game board out of construction paper that is shaped like a tree. Teachers first help the children cut a treetop and leaf shapes from green paper. They discuss children’s preferences for tall/short and thick/thin trunks, giving them practice using descriptive vocabulary words, particularly ones related to size. This step builds both general and math concept vocabulary. Children cut trunks from brown paper and paste/glue them on the treetops. While out of the children’s sight, the teacher cuts 5 to 10 (or more) pairs of shapes per child/tree from different colors of construction paper. Pairs should not match exactly (e.g., a blue square can be paired with a red square). The teacher glues one of each pair of shapes to each child’s tree while the child is not looking. The teacher then gives each child the rest of the shapes, and invites children to see how quickly they can match each shape to its “partner” on the tree. The teacher can provide “warmer/cooler” distance clues, and should provide reinforcement each time a child correctly matches a pair of shapes. Teachers can make this activity more challenging by using more shapes and/or getting students to match shapes that are different sizes (e.g., children can be asked to match smaller diamonds to larger diamonds).

Homemade Beanbag Game
Young children enjoy tossing objects and practicing their aim. Adults can make a beanbag game that helps children learn numbers and identify sets, while also allowing them to construct their own game rules. First, the adult should cover five big, equally-sized coffee (or similar) cans with paper that is adhesive on one side. The adult should then use markers to write a number from 1 to 5 and draw the corresponding number of dots on each can. The next step is to fill 15 tube socks with beans and knot/tie/sew them shut. The following numerals and the corresponding number of dots should be written on each homemade beanbag using markers: the number 1 on five beanbags, the number 2 on four beanbags, the number 3 on three beanbags, the number 4 on two beanbags, and the number 5 on one beanbag. Next, the adult should attach the cans to the floor with tape or Velcro. Then, the adult should mark a line on the floor that children must stand behind, and should direct children ONLY to toss the beanbags into the cans. Children will devise various games/rules. First, they may simply toss the beanbags into the cans; then, some may try to toss beanbags into a can that has the same number as the one marked on the beanbag. Eventually, some may throw three beanbags into the “3” can. They may/may not keep score. Allowing children to determine the details and rules gives them an opportunity to develop their imagination and decision making skills, and to create their own games while learning number and set identification.

Guessing Game
Adults can adapt the format of “20 Questions,” “I Spy,” and other similar guessing games to focus on numbers and help children learn number concepts. For example, adults could say, “I’m thinking of a number from 1 to 10….” and then give children 10 guesses. Adults give children cues as they guess, such as “higher” and “lower,” to help them narrow down the number of possible correct answers. As children improve, adults can increase the number range (e.g., from 0 to 50) or use larger numbers (e.g., from 20 to 40). As children’s skills and self-confidence develop, adults can reverse roles, having children think of numbers and give clues while adults guess. Young children enjoy the fun of guessing, getting closer using clues, deducing correct answers, and fooling adults with their own clues. Concurrently, they learn to describe numbers, compare them, and sequence them. Adults can make the game more difficult by limiting the number of guesses allowed and/or setting time limits. They can make it easier by providing a written number line for children to reference. This game requires no materials (or just a basic number line), is a great way to pass time, and entertains children while helping to develop numeracy skills.

Arts and Crafts
According to the U.S. Department of Agriculture, preschoolers need three ½-cup servings of fruit and three ½-cup servings of vegetables daily. However, many young children are picky/resistant. Adults can motivate them to eat produce with a “food rainbow” project. Adults show children a picture of a rainbow, and discuss its colors and their sequence (teaching some earth science, optics, and color theory!). A fun art project is allowing students to color their own rainbows, which improves fine motor skills. Then, adults can have children cut out pictures from grocery circulars and name each food. The adult can help children find one healthy fruit/vegetable for each color, gluing each food to its corresponding stripe on the rainbow. Adults can then help children pull apart cotton balls and glue them to their rainbow pictures to represent clouds. Children can then post their food rainbows on refrigerators as artwork and as healthy eating reminders. At the bottom, children can draw and color one box (bottom-up) for each food they eat (e.g., blue = blueberries, orange = carrots, red = apples, etc.) to create a bar graph. Children should try to “eat” the entire rainbow every week. This activity gives children the opportunity to produce colorful art, eat better, track and document their diets, and develop graphing skills.

Treasure Hunt
A treasure hunt is an ideal outdoor activity for young children, and can also be adapted for indoor fun. The treasure can be anything (e.g., a small toy/play money/chocolate “coins”/rocks spray painted gold or silver, etc.). The adult should put the treasure in a paper bag marked with a large X. The adult should hide it somewhere where it is not visible, but will not be overly difficult for children to find. Then, the adult should make a treasure map, using few words and many pictures, sketching landmark objects in the area (trees, houses, etc. if the activity will be done outdoors, and furniture, walls, etc. if the activity will be done indoors). The adult should ensure the map is developmentally appropriate for young children, and that they will be able to read it independently. Adults with time and motivation can make the map look authentic by soaking it in tea/coffee, drying it in a 200° oven, or even charring its edges. Adults should include a dotted line on the map that reinforces the simple directions and indicates the path to the treasure, which is indicated on the map by a large X. Children have fun, use their imaginations, make connections between symbols and images to corresponding real-world physical objects, and begin learning to read maps.

Pasta Necklace
Stringing beads/noodles is an activity that helps young children develop hand-eye coordination, which they will need for writing and other everyday activities that require fine motor coordination. Noodles are typically the perfect size for young children’s hands. They are inexpensive, usually costing less than comparably-sized beads. Moreover, pasta is non-toxic, an advantage when working with little persons who put things in their mouths. Hollow, tubular noodles like penne, ziti, wagon wheels, etc. are ideal. Fishing line/craft beading string/other stiff string is best; soft, limp string/yarn is harder for young children to manipulate. Using multicolored vegetable pasta removes the need to use markers or dye to add color. If using white pasta, children can color the noodles with markers, but adults should keep in mind that the ink can bleed onto skin/clothes even when it is dry. Adults should cut pieces of string that are long enough to allow children to easily slip the necklaces on and off after they are tied. Adults should also use a knot to secure a noodle to one end of the string. By providing more than one noodle shape, adults can invite children to string the noodles to create patterns, which develops pattern recognition and pattern creation abilities. These abilities also inform repetition, rhythm, categorization, and sequencing skills, which are important in math, music, art, literature, clothing design, etc.

Number Dash
A game for young children that some educators call “Number Dash” (Miller, ed. Charner, 2009) builds foundational math concepts and skills, while providing physical activity. It can involve small or large groups (the referenced authors say “the more the merrier”). Help children write large numbers on a paved area with sidewalk chalk. Make sure numbers are spread far enough apart so children will not collide while running. There should be one of each number for each child (e.g., six “1s,” “2s,” “3s,” etc. if there are six children). Use chalk colors that contrast with the pavement color to ensure the numbers will be highly visible. Tell children to run (“dash”) to whichever number you call out and stand on it until you call another number. Call out numbers randomly. Encourage children who have located the number to help their classmates/playmates. This game develops gross motor skills, number writing skills, and number recognition skills. It also provides experience with playing organized games, following rules, following directions, and cooperating with and helping others. This game can also be played with letters, colors, and/or shapes.

Introducing Standard Measurement Using a Ruler
A teacher is introducing standard measures to her class as part of a unit on measurement, one of the early math skills. She shows the children a ruler, explaining that it is one foot long, and that we can use it to measure inches and parts of inches. She demonstrates placing the ruler on paper to measure a given length, explaining that the ruler can also be used as a straight edge for drawing lines. One child asks, “How come you started with zero? Why don’t you start with one like when we count?” The teacher responds, “That’s a very good question! Zero means none/nothing. When we count, we start with one because we already have at least one of something. When you were born, you were not one year old; your age began at zero. After a year, on your first birthday, you were one year old. We also begin measuring distances at zero/none/nothing. The first piece/unit of measurement is one, not two. The distance from zero to one is equal to one. To get to one inch, for example, we need to start at zero.”

Learning About Geometric Shapes and Their Properties
A teacher has been working with students to help them develop their shape identification skills. They can recognize shapes by sight, and have also learned the defining properties of different shapes (number of sides, etc.). The teacher shows the class this figure:

She asks how many rectangles they can find in the figure. One student answers, “There is one rectangle,” which is incorrect because a square is a rectangle; this figure has four rectangles that are squares. Moreover, the entire figure is itself a rectangle. Another student therefore says, “There are five rectangles.” This response is also incorrect. Two adjacent squares also form a rectangle; this means there are three additional rectangles. Three adjacent squares also form a rectangle; this means there are two additional rectangles. Thus, the figure has a total of 10 rectangles. Solving this puzzle requires the use of many skills, including analyzing visual information, synthesizing visual information, recognizing patterns, recognizing shapes, and identifying the properties of shapes.

Collecting, Organizing, and Displaying Data Using Sticky Notes and a Teacher-Made Chart
A preschool teacher is teaching her group of ten children about basic data collection, data arrangement, and data display. She shows children yellow, blue, and green sticky notes, and has each child select his/her favorite color. Five children choose yellow notes, three select blue, and two choose green. By choosing one of three colors, each child has participated in data collection. The teacher draws lines to divide a sheet of paper into three columns, and labels each column with one of the colors. She helps the children place their chosen sticky notes in the correct columns. By arranging the colored sticky notes into columns, the teacher and children have organized the data they gathered. Once all notes are in their proper color columns, the completed chart is an example of how collected, organized data can be displayed.

Yellow sticky note

Blue sticky note

Green sticky note

Selecting One of Three Colors of Sticky Notes, Organizing Them by Color, and Displaying Them
The teacher had ten children each choose one of three colors of sticky notes, an example of basic data collection. She used a chart with three columns to organize the children’s choices as follows:

Yellow sticky note

Blue sticky note

Yellow sticky note

Green sticky note

The chart displays the collected and organized data. The teacher asks the children which color was chosen the most. Seeing five yellow notes, they answer, “yellow.” She asks which color was chosen the least, and they say, “green.” She asks them to use numbers to arrange the color choices from most popular to least popular. They arrive at, “five yellow, three blue, and two green.” Together, the teacher and the children point to and count ten children. She tells them five equals half of ten, and asks which color half of the children chose. Together, they figure out it was yellow. These are examples of analyzing and interpreting data.