Manuscript for Preschool Math Module 2: Number and Operations for Preschoolers Slide #1: Welcome to the preschool math module entitled “Number and Operations for Preschoolers.” In the previous module, we explored the importance of early childhood math, reviewed the recommendations of the National Council of Teachers of Mathematics (NCTM) and National Association for the Education of Young Children(NAEYC) regarding preschool math, explored the Texas Prekindergarten Guidelines for mathematics, examined the importance of having high expectations for all children regardless of gender, race, socioeconomic or language background, reflected on our own math anxiety and found ways to overcome it, and distinguished the components of a math‐rich learning environment for preschoolers. Slide #2: There are five objectives for this training. We will learn about the main components of number and operations, examine the complexity of counting, experience a variety of counting tasks for preschoolers, and learn developmentally appropriate approaches for teaching addition and subtraction. Slide #3: This module is divided into the following three sections: 1) Components of Number and Operations and Tasks that Develop Understanding of Number, 2) Counting Development, and 3) Methods for Assessing and Developing Preschoolers’ Understanding of Number. In several of the sections you will be asked to refer to Handout #2. This handout contains helpful references, additional information, engaging and worthwhile preschool math tasks and reflective activities to increase your involvement throughout the training module. While the powerpoint gives brief descriptions of engaging tasks for preschoolers, it is best to print Handout #2 so you have thorough descriptions of tasks including the materials and step‐by‐step instructions for using the tasks with your children. Slide #4: The first section of this training will focus on the components of number and operations. Number is more than just counting. It also includes developing a sense of numbers, how they are used, how they relate to one another, and how to perform operations such as addition and subtraction. This section will also provide many rich and worthwhile mathematical tasks that are appropriate for preschoolers. I hope these tasks will open your eyes to the wonder of number. Many of the activities can be completed with very few materials and simple planning. But they contain powerful learning ideas that will help children build a firm foundation of number understanding. You will also notice many of the activities are designed for centers. Research tells us young children respond better to small‐group instruction (centers) than whole class lessons (Bredekamp, 2011). Centers allow children to get their hands on math materials and their minds on math concepts. Centers where children work with a small group of classmates also facilitate communication about math—an important process for mathematical growth and understanding. Manuscript for Preschool Math Module 2: Number and Operations for Preschoolers COMPLETE 9/14/2015 P a g e | 1 Manuscript for Preschool Math Module 2: Number and Operations for Preschoolers Slide #5: Toddlers and preschoolers encounter numbers daily. They celebrate a birthday, see numbers on a microwave oven display, count stairs to their apartment, and so on. Familiar nursery rhymes chime, “One, two, buckle my shoe” and “Yes, sir. Yes, sir. Three bags full. One for my master. One for the dame. And one for the little boy who lives in the lane.” Some of their favorite picture books are probably counting books. The Committee on Early Childhood Mathematics of the National Research Council recommends that number and operations should be given greater emphasis than secondary topics such as geometry and measurement (Cross, Woods, & Schweingruber, 2009). Even our Texas PreK guidelines communicate the importance of establishing strong foundations in number by including 12 objectives for number and only 11 objectives for the other three content areas in math (geometry, measurement, and algebra and data analysis) combined. Slide #6: When Howard Gardner said counting was a natural instinct for young children, he echoed research showing that number knowledge emerges surprisingly early in life and develops considerably during the preschool years (Clements, 2004). Number receives much of the attention of mathematics education researchers who seek to understand how children develop knowledge of number. For some examples of this research, see Baroody (1999, 2004), Fuson (1988), Kilpatrick, Swafford, and Findell (2001), and Kamii (1985). Slide #7: Numbers are part of our environment and essential to our navigation through daily life. Look at the seven uses for numbers listed on the slide. Which of these ideas are you already familiar with? Which terms are new to you? Rote counting means reciting numbers without necessarily placing a value on the number—just saying the sequence of numbers—and is an important first step in being a competent counter. Rational counting, on the other hand, involves counting objects with accuracy. Subitizing is an immediate apprehension of the number of items in a small set (usually 1‐4 items) without counting. Cardinal numbers are the counting numbers 1, 2, 3, etc. We use the cardinal numbers to show how many items are in a set. First, second, and third are examples of ordinal numbers. We report measurements in numbers of pounds, inches, liters, and so on. Finally, nominal numbers are sequences of numbers where the digits themselves have no relationship to quantity. Locker combinations, phone numbers, zip codes and pin numbers are nominal numbers. Slide #8: Some of the components of number and operations for young children include counting, adding to and taking away, composing and decomposing, and grouping and place value. These are the components we will focus on in this module. We’ll be giving extra attention to counting. Manuscript for Preschool Math Module 2: Number and Operations for Preschoolers COMPLETE 9/14/2015 P a g e | 2 Manuscript for Preschool Math Module 2: Number and Operations for Preschoolers Slide #9: Number composition and decomposition is a way of breaking apart whole numbers in ways that make them easier to work with. For instance, if you want to add 235 + 165, you could think: 235 is made up of 200 and 35. I can add the 35 to the 165 to make 200. Then I have 200 + 200 = 400. Easy! Of course, with preschool‐age children number composition and decomposition is a simpler process. For example, a preschooler can think of the number 4 as 1 and 3 or 2 and 2. Some researchers call these “break‐apart partners” Slide #10: One way to describe the smaller numbers that make up a larger number is to refer to them as partitions. Just like you might use bookcases to partition your classroom into small workstations, we can partition numbers. You might also hear partitions with just two addends referred to as “break‐apart partners.” Slide #11: Remember that one of the NCTM Process Standards is reasoning. We don’t want children to just memorize math facts and rote procedures. We want them to make sense of math and discover relationships between mathematical ideas. Number composition and decomposition are reasoning processes that help children to see relationships between numbers. They begin to conceptualize the ideas of “parts” and “wholes” which are fundamental to understanding addition and subtraction, fractions, and many other mathematical ideas. When children can take numbers apart in flexible ways, they can use mental strategies to solve computation problems instead of relying on time‐consuming paper and pencil approaches and algorithms. Our place value system (tens and ones) is essentially a number composition and decomposition concept in that we can think of 22 as two sets of ten and 2 ones. Slide #12: Children need ample hands‐on, meaningful experiences with number composition and decomposition. This powerpoint and Handout #2 give several engaging activities preschoolers will enjoy. As you read through the activities, think of ways you can adjust them for the children you teach. You may need to make the activities a bit more challenging for older children or a bit more teacher‐guided for younger children. These activities are inexpensive, require few materials, and can be easily implemented in your classroom. As you go through the activities, try to visualize how you will use them with your children. Slide #13: This activity builds hand‐eye coordination as well as fine motor control. The child investigates the composition of numbers by sliding the beads to separate them into break‐apart partners. As always, be sure that children do not swallow beads. Use only one color of beads for Manuscript for Preschool Math Module 2: Number and Operations for Preschoolers COMPLETE 9/14/2015 P a g e | 3 Manuscript for Preschool Math Module 2: Number and Operations for Preschoolers each bracelet. This will help eliminate the distraction that the color attribute can be. If several colors are used, children will focus so strongly on the attribute of color that they will not be able to see other partitions. You can make up an optional recording sheet if appropriate. The recording sheet shown on Handout #2 is a format that will work well for preschoolers. Slide #14 Break‐Apart Baggies is a simple task, but it offers strong conceptual exploration. Children slide the counters across the line to find all of the break‐apart partners that compose the target number. Again, the recording sheet is optional. Younger children can say the break‐
apart partners aloud to you or to another child in the center. This is also a good math activity to send home with each child. Send a short note explaining to families how the baggies contribute to children’s understanding of the composition of number. Explain how the counters are slid across the line to show the break‐apart partners. You might encourage parents to make their own set of baggies to use at home. Slide #15: Children enjoy the active nature of the Hoop Pom Pom Toss Game. The slide shows one possible way children may record their results on a half‐sheet of paper by drawing a large circle then coloring dots that land in and out of the circle. Or they may use the recording sheet for break‐apart partner activities. What other ways might children record the results of the activity? Watch the video of children playing the Hoop Game. While you’re watching notice how the teacher facilitates children’s understanding of the number five? Write your reflection on Handout #2. Slide #16: Bears in a Cave is a classic number composition and decomposition task for young children. It gives another context for helping children visualize the missing part of a whole set of items. If you don’t have bear counters, the task can easily be adapted as Cars in a Garage (use matchbox cards and a box) or Frogs in a Log (use small frog counters and a paper towel tube). The mathematical communication that children engage in during the task is very valuable in building vocabulary such as take away¸ all together, some, more, less, and fewer. What other vocabulary might be used in the task? Slide #17: This is an early subtraction or missing addend activity. Children think about the composition of the number and supply the missing break‐apart partner. For example, if you Manuscript for Preschool Math Module 2: Number and Operations for Preschoolers COMPLETE 9/14/2015 P a g e | 4 Manuscript for Preschool Math Module 2: Number and Operations for Preschoolers show a plate with three dots, you would say, “I have three. I wish I had six.” Children could then use a variety of strategies to count up from three to six. They might use blocks, their fingers, or mental math to find the missing part—three. This is a powerful task for helping children begin to count with a number other than one. Slide #18: This activity helps children to manipulate the break‐apart partners for numbers. Using a cute animal plate is optional, of course. The idea is to help children break the whole number into number bonds or break‐apart partners in lots of different ways. Younger children might count and say the break‐apart partners aloud to you or to a friend in the center. Older children can use a recording format for break‐apart partners like the one shown on Handout #2. Slide #19: If you don’t already have a balance scale in your classroom, it is highly recommended that you purchase one, if possible. It can be used in the math center or the science center to help explore concepts of equality as well as weight comparisons. The balance scale is typically used for comparing the weights of two objects. For Balance Scale Exploration, however, guide children to use just one type of counter—unifix cubes, for example—and make the two sides equally balanced. Use mathematical vocabulary such as added, taken away, equal, same, and so on. Use other counters such as frogs, dinosaurs, etc. continuing to emphasize how many counters must be added or taken away to make the scale equally balanced. Slide #20: Put a star by the Number Composition and Decomposition tasks you are eager to try with your children. Make a list of items you will need to purchase or borrow to incorporate the tasks into your classroom. Remember that if a task seems too hard or too easy for your children, you can adapt it to meet their needs. Be creative and strive to make math meaningful for your children. Slide #21: The following slides offer a few hands‐on, engaging tasks for practicing grouping. As you read through the slides and Handout #2, identify which of the tasks you would like to use with your children. Put a star by those tasks on your handout. Slide #22: Grouping is the process of combining objects into sets with each having the same number of objects. Early experiences with grouping objects help lay a foundation for our base‐
ten system of ones and tens. Skip counting, multiplication and division, and measuring with standard units all involve grouping. Manuscript for Preschool Math Module 2: Number and Operations for Preschoolers COMPLETE 9/14/2015 P a g e | 5 Manuscript for Preschool Math Module 2: Number and Operations for Preschoolers You can provide your children hands‐on experience with grouping by:  Giving them a small bowl of counters and asking, “I wonder how many sets of five we can make out of these counters?” Then have them count out sets of five items and “leftovers.” This is a great activity to do in pairs so that children can talk about the math involved. Repeat the activity with sets of ten.  You can also provide your children hands‐on experience with grouping by giving children blank 5‐ or 10‐frames and a bowl of counters. Repeat the activity above. For older children, you may ask them to count by 5s or 10s (depending on if this has been introduced previously). Slide #23: Children have a lot of fun counting and grouping as they play Make Four Elbows. It’s a quick activity that you can use in those few moments before your class must leave for the playground or lunch. Make Four Elbows also works well as a circle‐time activity. It supports children’s physical and social development as well as cognitive development. Of course, children love to be physically and mentally active. Slide #24: Working with and around the number ten is an important foundational skill for your children. Ten is a landmark or benchmark number. It is essential that children have ample experiences with the number ten. For the Make Ten Sticker Activity, prepare several sets of cards in advance. You may want to laminate these cards so that they can be used multiple times. A child completes a set of ten by adding stickers to those you have already placed on the card. It is not necessary for children to write the complete number sentence (for example 3+7=10). But you may ask older children to use the recording format from the Number Composition and Decomposition tasks to record their findings. Slide #25: Wrapping Paper Sets gives children experience with grouping objects. Ask parents to donate scraps of patterned wrapping paper. Any type will do as long as it shows objects that can be easily grouped by circling with a marker. Begin by having children circle sets of two, then move on to three, four, and five using different sheets of wrapping paper. Slide #26: You may be familiar with a 2x5 grid commonly called a ten frame. The ten frame is a visual recognition tool that helps children see a number’s relationship to ten. To fill a ten frame, have the child place counters top‐to‐bottom and left‐to‐right. This tool helps children to perform mental calculation using the benchmark numbers five and ten. Ten frames also help children begin counting from a number other than one. For example, if a child knows there are five filled spaces on the top row, they can start counting from five instead of one. Manuscript for Preschool Math Module 2: Number and Operations for Preschoolers COMPLETE 9/14/2015 P a g e | 6 Manuscript for Preschool Math Module 2: Number and Operations for Preschoolers Handout #2 provides a set of ten frame cards. You can use these cards for the Ten Frame Flash game shown on slide #30. They can also be used in any game that calls for dice. It is recommended that you print these cards on heavy‐weight paper and laminate them. You may even choose to enlarge them on a copy machine to make it easier for children to see and count the dots. Slide #27: Ten Frame Flash is a great sponge activity when you have just a few moments to fill. Or it may be used in a teacher‐led math center or whole group situation. It helps children to quickly recognize number patterns on a ten frame.  Show a ten‐frame card and see how fast the children can tell how many dots are shown.  For a variation, ask how many spaces are unfilled.  For a more difficult task, ask children to tell you one more than the number of dots shown.  This activity is fast‐paced and can be done in a few minutes at any time during the day. Slide #28: In Ten Frame Fill, children use a die to generate the first set of red counters on their ten frame. Then they guess how many more yellow counters they’ll need to fill the frame. After they guess, they use yellow counters to fill the frame and check their guess. You may ask older children to record the break‐apart partners for ten. Challenge them to find all six possible combinations of numbers that fill the frame. For a variation, use a ten‐sided die. This will allow children to find all the possible combinations of numbers that fill the frame. Slide #29: The National Council of Teachers of Mathematics offers a free online activity called Ten Frame. It provides four different child‐friendly tasks. First, children can report how many dots are in the frame. This task is good for matching numerals to quantities and is similar to the Ten Frame Flash activity. Second, children can follow the computer’s verbal instructions to fill a frame. This is good for one‐to‐one correspondence and counting. In the third activity, children view a frame that is partially filled and then click on counters to completely fill the frame. This is similar to the Ten Frame Fill activity. Fourth, children can use the ten frame to practice addition. If desired, the teacher may select a random ordering and give children practice in all of the activities. Explore this online game and consider ways you might use technology to encourage children’s interest in mathematics. Fill out your reflections on Handout #2. Manuscript for Preschool Math Module 2: Number and Operations for Preschoolers COMPLETE 9/14/2015 P a g e | 7 Manuscript for Preschool Math Module 2: Number and Operations for Preschoolers Slide #30: It is hoped you’ve enjoyed learning some engaging grouping and place value activities for your children. Take a moment to review the Grouping and Place Value activities listed in Handout #2. Which tasks are you eager to try out right away? Put a star by them. Make a list of materials you’ll need to purchase or borrow for the tasks. Slide #31: Are four‐year‐olds ready for addition and subtraction? Sure, if it’s experienced in a meaningful context. Let’s find out what types of activities are developmentally appropriate for teaching addition and subtraction to young children. Slide #32: Young children may not be ready to interpret the addition number sentence (For example: 2+3=5), but they are ready to talk about and model addition in contexts that are meaningful to them. Subtraction, likewise, is a natural part of young children’s daily experience. They know about eating snacks and having them go away, about toys being lost, and so forth. Do not rush too quickly to the written number sentence. Instead, focus on building the concept of addition as joining two sets and subtraction as take away. For example, sing songs with addition and take away scenarios. Act them out with finger puppets and counters. Keep addition and take away activities concrete by using hands‐on materials. The activities, games, and tasks in this section will help you see ways to make operations on number meaningful to preschoolers. Slide #33: The Disappearing Train gives experience with counting, subtracting, and comparisons between sets. Children work with a partner, which also encourages mathematical communication. Try a variation of the game where the winner is the player who collects the most cubes. Or try using a nontraditional dice or spinner that gives directions to add or subtract cubes. What strategies do you think children will use to determine the winner? Slide #34: Adding Up the Beats gives children a chance to count using their auditory sense rather than their sense of touch. Integrating math with music and movement also helps to develop the cognitive and physical domains simultaneously. This activity really grabs children’s attention. It would work well in a station where the teacher works with a small group of 4‐5 children at a time. To do this activity:  Hit a drum and have children show the number of beats by raising the fingers of one hand,  Hit the drum again and have them show the number of beats on the other hand,  Ask the children to add the number of beats shown on each hand and tell the total number of beats in all, Manuscript for Preschool Math Module 2: Number and Operations for Preschoolers COMPLETE 9/14/2015 P a g e | 8 Manuscript for Preschool Math Module 2: Number and Operations for Preschoolers 
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Place a set of counters in the middle of the table. Choose a child to strike the drum and have the rest of the group use the counters to show the number of beats. Join the sets to show the number of beats in all, and Show children a number sentence. Have children clap to show the beats from the number sentence. Then have them model the number sentence with counters, joining their two sets of counters (from the two sets of beats) to model the process of addition. Slide #35: Domino Parking Lot integrates counting, addition, and comparison between sets. Handout #2 provides a blackline master of the domino parking lot mat. You may want to print the mats on cardstock and laminate them for durability. To do this activity: • Place dominoes face down on the table. • Children take turns drawing a domino, counting (or adding) the number of dots on both sides of the domino and placing it in the correct “parking spot” on the mat. • Each person takes ten turns. • At the end of ten turns, the person with the tallest stack on any parking spot is the winner. If children have difficulty keeping track of their ten turns, use only 20 dominoes in the game. When the dominoes are gone, the game is over. What strategies do you think the children will use to determine a winner? Slide #36: Remember preschoolers are not ready for stacks of worksheets with addition or subtraction problems to solve. In fact, such practices are counter‐productive for building the concepts behind the operations. Instead, spend time talking about addition and subtraction in the real world modeling with real objects. Ask good questions and keep the process informal. Slide #37: Point out the addition and subtraction children use during their play. During center time, scaffold their learning by drawing children’s attention to joining and separating that they naturally do during play. For example, when a child is building a block tower, ask her to count the blocks then ask, “How many blocks would you have if you added two more?” Also, use familiar stories. For example, you could ask, ““If three little kittens each lost one mitten, how many would they have left?” Addition and subtraction are definitely already taking place all over your classroom. Manuscript for Preschool Math Module 2: Number and Operations for Preschoolers COMPLETE 9/14/2015 P a g e | 9 Manuscript for Preschool Math Module 2: Number and Operations for Preschoolers Slide #38: Because number concepts are the gateway to all other mathematical content, children must become competent counters. Counting may seem simple, but it is actually a highly cognitive endeavor. Let’s find out more about counting as a developmental process. Slide #39: Gelman and Gallistel’s research from the 1970s is considered foundational to all counting research over the past 40 years. They identified five essential elements of counting that must be developed before a child can be considered a competent counter. The stable order principle means the child understands that the sequence of numbers stays the same no matter what is counted. One‐to‐one correspondence is demonstrated when a child says one and only one number word for each item counted. Cardinality means the child recognizes that the final number spoken is the quantity in the set. A child who has developed order irrelevance knows that objects in a set can be counted in any order and it won’t affect the number of items in the set. Finally, abstraction refers to the understanding that anything or parts of things can be counted. Let’s find out how you, as a teacher, can help emergent counters to develop these essential understandings. Slide #40: Rote counting is simply repeating the counting sequence in order. A child who can rote count may not necessarily be able to count with meaning, but rote counting is an important milestone in learning to count with accuracy. Teaching a child the number sequence can be challenging. Try these ideas to keep children engaged: 1) Have children sit in a circle and toss or roll a ball. Count each toss or roll, stopping when you reach your target number. Then repeat. 2) Have a struggling counter sit on the floor across from a partner who is a competent counter. Blow up a balloon and have the children count each time someone hits the balloon. Start over when the balloon hits the ground. 3) Count aloud as children repeat a series of motions such as clapping, hopping, or bending their knees. Counting shouldn’t be isolated to calendar time or the math center. Instead, incorporate counting into the daily routines of the classroom by counting whenever possible. When children see counting as a useful skill, they will be more motivated to learn to do it themselves. Manuscript for Preschool Math Module 2: Number and Operations for Preschoolers COMPLETE 9/14/2015 P a g e | 10 Manuscript for Preschool Math Module 2: Number and Operations for Preschoolers Slide #41: Beyond knowing the number words in sequence, a child must make the connection between touching one‐and‐only‐one object and saying a single number word. Emergent counters may simply recite the counting sequence, touching more than one object for each number said or by touching one item and saying several number words. This is especially common when a set of objects contains more than five or six items to be counted. A counting center is a fantastic way to help build competence with 1‐1 correspondence. Early on, use only small sets of five or fewer objects. Children can make one‐to‐one correspondences with pairs of objects as well. For instance, children can put a blue bear on a blue tile, a red bear on a red tile, etc. Nuts and bolts or cups and plates make good tools for understanding one‐to‐
one correspondence. Observe a child carefully to see if he or she speaks one number word for each item touched. If not, ask the child to count slowly and touch each object. You may want to place your hand gently on top of the child’s and touch and count with him. Then have the child repeat the process without your help. Another strategy is to have the child actually pick up the items and move them once they’ve been counted. Slide #42: Cardinality means the child understands that the last number said is the number of items in the set. Read through the Cardinality Scenario on page 5 of Handout #2. While the child has mastered the counting sequence to the number five, he does not yet recognize that the five he says at the end of the sequence indicates the total number in the set. Observe your children carefully to see which of them have developed a sense of cardinality and which have not yet mastered this concept. To facilitate cardinality, model the concept by saying, “So you have five,” when the child finishes counting five objects. Have the child count another set of objects (three, for example) and ask, “How many do you have?” If the child begins to count the entire set, supply the total again by saying, “So you have three.” Continue this process over time to help the child see the relationship between the last number said and the number of items in the set. Slide #43: Emergent counters sometimes wonder if the order in which objects are counted somehow affects the number of objects in the set. Children need ample practice with order irrelevance to see that objects can be counted in any direction (top‐to‐bottom or bottom‐to‐
top, left‐to‐right or right‐to‐left), and it does not affect the number of objects. Children may be engaged when they get to use a counting stick (such as a decorated ruler) to point to objects in various orders. Or, ask children to line up in a certain order, count the children, then have them mix up and count them again. Ask why the number of children in the class did not change. Manuscript for Preschool Math Module 2: Number and Operations for Preschoolers COMPLETE 9/14/2015 P a g e | 11 Manuscript for Preschool Math Module 2: Number and Operations for Preschoolers Slide #44: Some children come to us having experienced counting in their homes. Their parents may have encouraged them to count steps to their apartment, to count cans as they’re placed in the shopping cart, or to count cars in the parking lot. Other children come to us with very little experience with counting. These children need to see they can count anything and that parts of things can be counted as well. When putting together a farm scene puzzle, count the pieces, count the number of legs on the cows, count the number of ears on the sheep, etc. to model the abstraction principle of counting. Children also love to use their own bodies as manipulatives for counting. You can count girls and boys. Then count the number of girls’ eyes, boys’ ears, etc. Remember, you are demonstrating that anything can be counted and that parts of things can be counted. Slide #45: Remember when we discussed earlier that counting must be important because there are so many more PreK Guidelines for counting than for the other mathematics? Look over the summary of the objectives for counting once again. These are the concepts the state of Texas hopes children will have mastered before entering kindergarten. Which of these counting milestones do you feel your children are ready to learn, have learned already, or need more practice with? We’ve discussed most of the objectives already. But, you may not be familiar with two of the counting objectives. Let’s learn a bit about subitizing and ordinal numbers. Slide #46: The term subitizing is derived from the Latin adjective subitus, meaning sudden, and the Latin verb subitare, meaning to arrive suddenly. Though subitizing is just beginning to appear in the mathematics curriculum, the phenomenon has been studied for over 100 years. A researcher named Jevons (1871) may have been the first to test the subitizing phenomenon in a passably scholarly experiment. Jevons grabbed a handful of beans and threw them into a box, glanced at them briefly, and estimated the number of beans. He then counted the beans to check. After completing 1,027 trials, Jevons noted that up to four objects could be estimated with complete accuracy, and up to five were estimated with very few errors. Slide #47: Subitizing may be an innate ability that all of us are born with, but it can also be developed with practice. To help children subitize small sets of numbers quickly, you can show various spatial arrangements for numbers such as dice, dominoes, dot plates, and ten frames. Flash these arrangements quickly to discourage counting and help children learn to recognize patterns in those arrangements. Lining up objects in a long line may discourage subitizing, but placing items in a three‐in‐a‐row arrangement may help encourage children’s subitizing and counting on from three. Finally, model subitizing during clean up time. When children are picking up blocks, join them and say, “Ah, hah. This time I grabbed three. This time I grabbed Manuscript for Preschool Math Module 2: Number and Operations for Preschoolers COMPLETE 9/14/2015 P a g e | 12 Manuscript for Preschool Math Module 2: Number and Operations for Preschoolers two.” Children will soon be joining in the subitizing fun as they pick up and “see” small sets of items. Slide #48: The PreK Guidelines include an objective for ordinal numbers. Ordinals such as first, second, third, and so on help children to see how numbers relate to one another in magnitude or position. Several picture books offer engaging examples of ordinals. Henry the Fourth by Stuart J. Murphy introduces children to four dogs who compete in a dog show. Pancakes for Breakfast is a wordless picture book by Tomie DePaola that could spark a conversation about the process of making pancakes. What ingredients do we add first, second, etc.? To give children meaningful experiences with ordinals, use ordinal numbers to describe where children are in line. “Ana is first. Sam is second. Who is third?” Make an attendance chart where children place their name card in the order they arrived at school. Have a class discussion about who arrived first, second, third, and so on. Centers can also give children experience with ordinals. For one center, place 5 crayons of different colors in a row. Ask child to identify which is first, second, etc. For another center, bring in a set of stuffed animals. Have children make a line of the stuffed animals and pretend it’s a parade. Ask children to point to the first animal in line, the fourth, the third, etc. Slide #49: While we shouldn’t endlessly drill to give children practice with recognizing the digits, we should use the digits in meaningful ways and throughout the classroom to expose children to them. Look through the list of ideas for Facilitating Recognition of Digits 0 through 9 in Handout #2. Which of these ideas are you already using in your classroom to help children recognize the digits? Star the ideas you would like to try. Slide #50: Learning to count with competence is a complex process requiring meaningful practice. The following activities include some whole‐group tasks and some tasks that are better for small groups of children to complete in a center. Children can complete some of the activities independently while others will require teacher facilitation. As you read through the activities, try to picture your class completing the activities. How will you arrange your schedule, materials, and environment to make these counting activities successful for your children? Slide #51: Counting is not magical. It makes sense, is consistent and logical, and follows a predicable sequence. Therefore, resist calling your counting stick a wand. Instead decorate a ruler, wooden dowel, or some other pointer to make it your class counting stick. PreK Guideline Manuscript for Preschool Math Module 2: Number and Operations for Preschoolers COMPLETE 9/14/2015 P a g e | 13 Manuscript for Preschool Math Module 2: Number and Operations for Preschoolers V.A.1 states children need to recognize that things (or parts of things) can be counted. A counting stick is an engaging way of drawing attention to things that can be counted. Slide #52: Snowball Hunt gets children up and moving as they search for hidden “snowballs” (cotton balls) throughout the classroom. As children find “snowballs,” they place them in the egg carton in the order of the counting sequence. This is an activity that would work well for whole or small group. Hiding the snowballs for the next set of hunters is a favorite, too. Slide #53: The concept of zero is somewhat illusive for children. They don’t often hear the word zero and can have a difficult time articulating what zero means. Playing the Zero Game as a whole class gives meaningful experiences with the concept of zero. Slide #54: In the Hop to It Game, children move their bodies as they identify the digits 0 through 9, thus addressing PreK Guideline V.A.9. Anytime you can encourage children to move to learn math, you’re creating a developmentally appropriate environment for learning. So get your class up and moving! Slide #55: Pair of Socks gives children experience with several early childhood objectives. Children sort objects and describe how groups are similar or different (V.E.1), count with one‐
to‐one correspondence (V.A.3), develop fine motor control as they squeeze a clothespin, and hone hand‐eye coordination as they pin socks to the clothesline. This is an activity that works best in a center. Slide #56: Most children love pizza and the idea of creating one’s own pizza is appealing and fun. Pizza, Pizza would work well in a center. Children recognize the digit on the card (V.A.9) then count with one‐to‐one correspondence (V.A.3) to place the correct number of pepperoni slices on their pizza. Completed paper pizzas can be displayed in the dramatic play center Pizza Shop or hung on a bulletin board. Remember that displaying children’s math work shows you value mathematics learning in your classroom. Or you may choose to do this activity for a snack and use English muffins and real pepperoni. For a more challenging variation, you may create a spinner showing different pizza toppings (pepperoni, olives, peppers, etc.) and have the child spin the spinner and roll a dice. The spinner shows the type of topping and the dice shows how many of the topping to add to the pizza. Manuscript for Preschool Math Module 2: Number and Operations for Preschoolers COMPLETE 9/14/2015 P a g e | 14 Manuscript for Preschool Math Module 2: Number and Operations for Preschoolers Slide #57: Flower Pot Counting is a hands‐on center activity where children recognize digits (V.A.9), count with one‐to‐one correspondence (V.A.3), and recognize that things or parts of things can be counted (V.A.1). Flowers and pots are available for purchase at discount stores. Slide #58: Fishy Bubble Counting is a math center activity that gives children practice ordering the numbers and matching a quantity to a number. First, children put the fish cutouts in order. Then, children match the number of flat‐sided marbles to the number written on the fish, placing the marbles above the fish as if they were bubbles coming out of the fish’s mouth. If your children are recording their learning on paper, they may make a journal entry to record their work. You might have them draw a picture showing a fish labeled with a number and with the correct number of bubbles coming out of its mouth. Slide #59: Read through the Tips for Teaching Counting on Handout #2. You probably already do several of the things on this list. Put a check mark next to those teaching practices. Which would you like to add to your teaching repertoire or classroom routine? Put a star next to those things you’d like to start doing with your children. It may take some focused energy to begin seeing and taking advantage of opportunities to help children count in meaningful ways, but it is worth it! Slide #60: Incidental learning is not enough to lay a strong foundation in counting. Teachers must plan focused time for learning in small groups and whole class. When we “mathematize” children’s everyday encounters and plan many rich math experiences, we build children’s abilities to count with meaning. At the same time, we must be sure to keep our expectations and activities developmentally appropriate. Math time should be a time of joy, exploration, and fun. Slide #61: Now that you know about the PreK Guidelines for counting and many tasks for helping children to become competent counters, you’re ready to look carefully at a child’s developmental stage of counting and make instructional decisions based on each child’s individual needs. Slide #62: Understanding children’s mathematical development helps teachers be knowledgeable and effective in teaching math. Children’s thinking follows natural developmental paths in learning math. When teachers understand these paths and offer Manuscript for Preschool Math Module 2: Number and Operations for Preschoolers COMPLETE 9/14/2015 P a g e | 15 Manuscript for Preschool Math Module 2: Number and Operations for Preschoolers activities based on children’s progress along them, they build math learning environments that are developmentally appropriate and particularly effective (Clements & Sarama, 2009). Counting is similar, so taking time to assess children’s develop in counting helps you to plan activities and support each child’s growing competence. Slide #66: There are several different methods for analyzing children’s development of counting. The one shown on this slide comes from Clements and Sarama (2009) and is in Handout 2. This system is based on children’s ages and levels of development. Notice that the children’s ages are given as a range and that one age may include several stages of counting development. Children’s counting can be thought of as progressing along a continuum of ever‐
increasing competence. Slide #67: This video shows three children at various stages of counting development. As you watch the video, notice how the teacher assesses different elements of counting—counting sequence, cardinality, one‐to‐one correspondence, etc. Use the handout to try to place each child in one of the stages of counting development. Slide #68: Let’s review the major ideas from this module. 1. The main components of number and operations for preschoolers are:  Counting,  Number composition and decomposition,  Grouping and place value, and  Adding and taking away. 2. The principles of counting include:  Stable order principle (the counting sequence stays the same no matter what is counted),  Correspondence (one number word for each object counted),  Cardinality (the last number said is the number in the set),  Order irrelevance (items can be counted in any order and the number in the sets stays the same), and  Abstraction (anything can be counted or parts of things can be counted). 3. The PreK Guidelines for number include the ideas listed in #2 above as well as recognizing the digits, subitizing, ordinal numbers, and meaningful addition and subtraction. Manuscript for Preschool Math Module 2: Number and Operations for Preschoolers COMPLETE 9/14/2015 P a g e | 16 Manuscript for Preschool Math Module 2: Number and Operations for Preschoolers 4. Assessing a child’s developmental stage of counting helps teachers to plan counting activities. 5. Counting can take place throughout the day and throughout the classroom, and 6. Children need planned activities for developing counting competence. Slide #66: Remember the pre‐test you took at the beginning of the module? Now it’s time for you to show what you’ve learned. Please click on the “Next Section” button at the bottom of this page to proceed to the post‐test. If you would like to review any of the previous sections, click on the desired title in the left menu bar. Good luck! Manuscript for Preschool Math Module 2: Number and Operations for Preschoolers COMPLETE 9/14/2015 P a g e | 17