5
The Teaching-Learning Paths for Number, Relations, and Operations

In this chapter we describe the teaching-learning paths for number, relations, and operations at each of the four age/grade steps (2- and 3-year-olds, 4-year-olds [prekindergarten], kindergarten, and Grade 1). As noted, the four steps are convenient age groupings, although, in fact, children’s development is continuous. There is considerable variability in the age at which children do particular numerical tasks (see the reviews of the literature in Clements and Sarama, 2007, 2008; Fuson, 1992a, 1992b; also see Chapter 4). However, a considerable amount of this variability comes from differences in the opportunities to learn these tasks and the opportunity to practice them with occasional feedback to correct errors and extend the learning. Once started along these numerical learning paths, children become interested in consolidating and extending their knowledge, practicing by themselves and seeking out additional information by asking questions and giving themselves new tasks. Home, child care, and preschool and school environments need to support children in this process of becoming a self-initiating and self-guiding learner and facilitate the carrying out of such learning. Targeted learning path time is also needed—time at home or in an early childhood learning center—that will support children in consolidating thinking at one step and moving along the learning path to the next step.

Although we consider the mathematics goals described in this and the next chapter foundational and achievable for all children in the designated age range for that step, we recognize that some children’s learning will be advanced while others’ functioning will be significantly behind. Children at particular ages/grades may be able to work correctly with larger numbers or more complex geometric ideas than those we specify in the various tables



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5 The Teaching-Learning Paths for Number, Relations, and Operations In this chapter we describe the teaching-learning paths for number, relations, and operations at each of the four age/grade steps (2- and 3-year- olds, 4-year-olds [prekindergarten], kindergarten, and Grade 1). As noted, the four steps are convenient age groupings, although, in fact, children’s development is continuous. There is considerable variability in the age at which children do particular numerical tasks (see the reviews of the litera- ture in Clements and Sarama, 2007, 2008; Fuson, 1992a, 1992b; also see Chapter 4). However, a considerable amount of this variability comes from differences in the opportunities to learn these tasks and the opportunity to practice them with occasional feedback to correct errors and extend the learning. Once started along these numerical learning paths, children be- come interested in consolidating and extending their knowledge, practicing by themselves and seeking out additional information by asking questions and giving themselves new tasks. Home, child care, and preschool and school environments need to support children in this process of becoming a self-initiating and self-guiding learner and facilitate the carrying out of such learning. Targeted learning path time is also needed—time at home or in an early childhood learning center—that will support children in consolidating thinking at one step and moving along the learning path to the next step. Although we consider the mathematics goals described in this and the next chapter foundational and achievable for all children in the designated age range for that step, we recognize that some children’s learning will be advanced while others’ functioning will be significantly behind. Children at particular ages/grades may be able to work correctly with larger numbers or more complex geometric ideas than those we specify in the various tables 12

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12 MATHEMATICS LEARNING IN EARLY CHILDHOOD and text. Each subsequent step assumes that children have had sufficient experiences with the topics in the previous step to learn the earlier content well. (See Box 5-1 for a discussion of what it means to learn something well.) However, many children can still learn the content at a given step without having fully mastered the previous content if they have sufficient time to learn and practice the more challenging content. Of course, some children have difficulty in learning certain kinds of mathematical concepts, and a few have really significant difficulties. But most children are capable of learning the foundational and achievable mathematics content specified in the learning steps outlined here. In both the number and operations and the geometry and measurement core areas, children learn about the basic numerical or geometric con- cepts and objects (numbers, shapes), and they also relate those objects and compose/decompose (operate on) them. Therefore, each core area begins by discussing the basic objects and then moves to the relations and operations on them. In all of these, it is important to consider how children perceive, say, describe/discuss, and construct these objects, relations, and operations. The development of the elements of the number core across ages is de- scribed first, and then the development of the relations and operations core BOX 5-1 Learning Something Well In most aspects of the number and the relations/operation core, children need a great deal of practice doing a task, even after they can do it correctly. The rea- sons for this vary a bit across different aspects, and no single word adequately captures this need, because the possible words often have somewhat different meanings for different people. Overlearning can capture this meaning, but it is not a common word and might be taken to mean something learned beyond what is necessary rather than something learned beyond the initial level of correctness. Automaticity is a word with technical meaning in some psychological literature as meaning a level of performance at which one can also do something else. But to some people it carries only a sense of rote performance. Fluency is the term used by several previous committees, and we have therefore chosen to continue this usage. Flu- ency also carries for some a connotation of flexibility because a person knows something well enough to use it adaptively. We find this meaning useful as well as the usual meaning of doing something rapidly and relatively effortlessly. Re- search on reading in early childhood has recently used fluency only in the latter sense as measured by performance on standardized tests of reading, such as the Dynamic Indicators of Basic Early Literacy Skills (DIBELS). We do not mean fluency to be restricted to this rote sense. By fluent we mean accurate and (fairly) rapid and (relatively) effortlessly with a basis of understanding that can support flexible performance when needed.

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12 PATHS FOR NUMBER, RELATIONS, AND OPERATIONS is summarized. These cores are quite related, and their relationships are discussed. Box 5-2 summarizes the steps along the teaching-learning paths in the core areas. As children move from age 2 through kindergarten, they learn to work with larger and more complicated numbers, make connec- tions across the mathematical contents of the core areas, learn more com- plex strategies, and move from working only with objects to using mental representations. This journey is full of interesting discoveries and patterns that can be supported at home and at care and education centers. THE NUMBER CORE The four mathematical aspects of the number core identified in Chap- ter 2 involve culturally specific ways that children learn to perceive, say, describe/discuss, and construct numbers. These involve 1. Cardinality: Children’s knowledge of cardinality (how many are in a set) increases as they learn specific number words for sets of objects they see (I want two crackers). 2. Number word list: Children begin to learn the ordered list of number words as a sort of chant separate from any use of that list in count- ing objects. 3. 1-to-1 counting correspondences: When children do begin counting, they must use one-to-one counting correspondences so that each object is paired with exactly one number word. 4. Written number symbols: Children learn written number symbols through having such symbols around them named by their number word (That is a two). Initially these four aspects are separate, and then children make vital con- nections. They first connect saying the number word list with 1-to-1 cor- respondences to begin counting objects. Initially this counting is just an activity without an understanding of the total amount (cardinality). If asked the question How many are there? after counting, children may count again (repeatedly) or give a number word different from the last counted word. Connecting counting and cardinality is a milestone in children’s numerical learning path that coordinates the first three aspects of the number core. As noted, we divide the teaching-learning path into four broad steps. In Step 1, for 2- and 3-year-olds, children learn about the separate aspects of number and then begin to coordinate them. In Step 2, for approximately 4-year-olds/prekindergartners, children extend their understanding to larger numbers. In Step 3, for approximately 5-year-olds/kindergartners, children integrate the aspects of number and begin to use a ten and some ones in teen numbers. In Step 4, approximately Grade 1, children see, count, write, and work with tens-units and ones-units from 1 to at least 100.

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10 MATHEMATICS LEARNING IN EARLY CHILDHOOD BOX 5-2 Overview of Steps in the Number, Relations, and Operations Core Steps in the Number Core Step 1 (ages 2 and 3): Beginning 2- and 3-year-olds learn the number core components for very small numbers: cardinality, number word list, 1-1 counting correspondences, and written number symbols; later 2- and 3-year-olds coordi- nate these number core components to count n things and, later, say the number counted. Step 2 (age 4/prekindergarten): Extend all four core components to larger numbers and also use conceptual subitizing if given learning opportunities to do so. Step 3 (age 5/kindergarten): Integrate all core components, see a ten and some ones in teen numbers, and relate ten ones to one ten and extend the core components to larger numbers. Step 4 (Grade 1): See, say, count, and write tens-units and ones-units from 1 to 100. Steps in the Relations (More Than/Less Than) Core Step 1 (ages 2 and 3): Use perceptual, length, and density strategies to find which is more for two numbers ≤ 5. Step 2 (age 4/prekindergarten): Use counting and matching strategies to find which is more (less) for two numbers ≤ 5. Step 3 (age 5/kindergarten): Kindergartners show comparing situation with objects or in a drawing and match or count to find out which is more and which is less for two numbers ≤ 10. Step 4 (Grade 1): Solve comparison word problems that ask, “How many more (less) is one group than another?” for two numbers ≤ 18. Steps in the Addition/Subtraction Operations Core Step 1 (ages 2 and 3): Use subitized and counted cardinality to solve situation and oral number word problems with totals ≤ 5; these are much easier to solve if objects present the situation rather than the child needing to present the situation and the solution. Step 2 (age 4/prekindergarten): Use conceptual subitizing and cardinal counting of objects or fingers to solve situation, word, and oral number word problems with totals ≤ 8. Step 3 (age 5/kindergarten): Use cardinal counting to solve situation, word, oral number word, and written numeral problems with totals ≤ 10. Step 4 (Grade 1): Use counting on solution procedures to solve all types of addition and subtraction word problems: Count on for problems with totals ≤ 18 and find subtraction as an unknown addend.

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11 PATHS FOR NUMBER, RELATIONS, AND OPERATIONS Step 1 (Ages 2-3) At this step, children first begin to learn the core components of num- ber: cardinality, the number word list, 1-to-1 correspondences, and written number symbols (see Box 5-3). BOX 5-3 Step 1 in the Number Core Children at particular ages/grades may exceed the specified numbers and be able to work correctly with larger numbers. The numbers for each age/grade are the foundational and achievable content for children at this age/grade. The major types of new learning for each age/grade are given in italics. Each level assumes that children have had sufficient learning experiences at the lower level to learn that content; many children can still learn the content at a level without having fully mastered the content at the lower level if they have sufficient time to learn and practice. Beginning 2- and 3-Year-Olds Learn the Number Core Components Cardinality: How many animals (crackers, fingers, circles, . . . )? uses perceptual subitizing to give the number for 1, 2, or 3 things. Number word list: Count as high as you can (no objects to count) says 1 to 6. 1-to-1 counting correspondences: Count these animals (crackers, fingers, circles, . . . ) or How many animals (crackers, fingers, circles, . . . )? counts ac- curately 1 to 3 things with 1-1 correspondence in time and in space. Written number symbols: This (2, 4, 1, etc.) is a______? knows some symbols; will vary. Later 2- and 3-Year-Olds Coordinate the Number Core Components Cardinality: Continues to generalize perceptual subitizing to new configurations and extends to some instances of conceptual subitizing for 4 and 5: can give number for 1 to 5 things. Number word list: Continues to extend and may be working on the irregular teen patterns and the early decade twenty to twenty-nine, etc., pattern: says 1 to 10. 1-to-1 counting correspondences: Continues to generalize to counting new things, including pictures, and to extend accurate correspondences to larger sets (accuracy will vary with effort): counts accurately 1 to 6 things. Written number symbols: Continues to learn new symbols if given such learning opportunities. Coordinates counting and cardinality into cardinal counting in which the last counted word tells how many and (also or later) tells the cardinality (the number in the set).

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12 MATHEMATICS LEARNING IN EARLY CHILDHOOD Cardinality The process of identifying the number of items in a small set (cardinal- ity) has been called subitizing. We will call it perceptual subitizing to differ- entiate it from the more advanced form we discuss later for larger numbers called conceptual subitizing (see Clements, 1999). For humans, the process of such verbal labeling can begin even before age 2 (see Chapter 3). It first involves objects that are physically present and then extends to nonpresent objects visualized mentally (for finer distinctions in this process, see Benson and Baroody, 2002). This is an extremely important conceptual step for attaching a number word to the perceived cardinality of the set. In fact, there is growing evidence that the number words are critical to toddlers’ construction of cardinal concepts of even small sets, like three and four and possibly one and two (Benson and Baroody, 2002; Spelke, 2003; also see Baroody, Lai, and Mix, 2006; and Mix, Sanhofer, and Baroody, 2005). Children generally learn the first 10 number words by rote first and do not recognize their relation to quantity (Fuson, 1988; Ginsburg, 1977; Lipton and Spelke, 2006; Wynn, 1990). They do, however, begin to learn sets of fingers that show small amounts (cardinalities). This is an important process, because these finger numbers will become tools for adding and subtracting (see research literature summarized in Clements and Sarama, 2007; Fuson, 1992a, 1992b). Interestingly, the conventions for counting on fingers vary across cultures (see Box 5-4). In order to fully understand cardinality, children need to be able to both generalize and extend the idea. That is, they need to generalize from a spe- cific example of two things (two crackers), to grasp the “two-ness” in any set of two things. They also need to extend their knowledge to larger and larger groups—from one and two to three, four, and five, although these are more difficult to see and label (Baroody, Lai, and Mix, 2006; Ginsburg, 1989). Children’s early notions of cardinality and how and when they learn to label small sets with number words are an active area of research at present. The timing of these insights seems to be related to the grammatical structure of the child’s native language (e.g., see the research summarized in Sarnecka et al., 2007). Later on, children can learn to quickly see the quantity in larger sets if these can be decomposed into smaller subitized numbers (e.g., I see two and three, and I know that makes fie). Following Clements (1999), we call such a process conceptual subitizing because it is based on visually appre- hending the pair of small numbers rather than on counting them. Concep- tual subitizing requires relating the two smaller numbers as addends within the conceptually subitized total. With experience, the move from seeing the smaller sets to seeing and knowing their total becomes so rapid that one can experience this as seeing 5 (rather than as seeing 2 and 3). Children may also learn particular patterns, such as the 5 pattern on a die. Because these

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1 PATHS FOR NUMBER, RELATIONS, AND OPERATIONS BOX 5-4 Using Fingers to Count: Cultural Differences Around the world, most children learn from their family one of the three major ways of raising (or in some cultures, lowering) fingers to show numbers. All of these methods can be seen in centers or schools with children coming from differ- ent parts of the world, as well as some less frequent methods (the Indian counting on cracks of fingers with the thumb, Japanese lowering and raising fingers). The most common way is to raise the thumb first and then the fingers in order across to the small finger. Another way is to raise the index finger, then the next fingers in order to the smallest finger, and then the thumb. The third way is to begin with the little finger and move across in order to the thumb. The first way is very frequent throughout Latin America, and the third way also is used by some children coming from Latin America. The second way is the most usual in the United States. It is the common way to show ages (for example, I am two years old by holding up the index and largest finger). This method allows children to hold down unused fingers with their thumb. But the other two methods show numbers in a regular pattern going across the fingers. Children in a center or school where children show numbers on fingers in different ways may come to use multiple methods. Because fingers are such an important tool for numerical problem solving, it is probably best not to force a child to change his or her method of showing numbers on fingers if it is well established. It is important for teachers to be aware and ac- cepting of these differences. kinds of patterns can also be considered in terms of addends that compose them, they are included in conceptual subitizing. Such patterns can help older children learn mathematically important groups, such as five and ten; these are discussed in the later levels and in the relations and operation core discussion of addition and subtraction composing/decomposing. Children also learn to assign a number to sets of entities they hear but do not see, such as drum beats or ringing bells. There is relatively little research on auditory quantities, and they play a much smaller role in ev- eryday life or in mathematics than do visual quantities. For these reasons, and because auditory quantities relate to music and rhythm and body move- ments, it seems sensible to have some activities in the classroom in which children repeat simple or complex sets they hear (clap clap or, later, clap clap clap pause clap clap), tell the number they hear (of bells, drumbeats, feet stamping, etc.), and produce sounds with body movements for particu- lar quantities (Let me hear three claps). In home and care/educational settings, it is important that early experi- ences with subitizing be provided with simple objects or pictures. Textbooks or worksheets often present sets that discourage subitizing and depict col- lections of objects that are difficult to count. Such complicating factors include embedded or overlapping pictures, complex noncompact things

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1 MATHEMATICS LEARNING IN EARLY CHILDHOOD or pictures (e.g., detailed animals of different sizes rather than circles or squares), lack of symmetry, and irregular arrangements (Clements and Sarama, 2007). The importance of facilitating subitizing is underscored by a series of studies, which first found that children’s spontaneous tendency to focus on numerosity was related to counting and arithmetic skills, then showed that it is possible to enhance such spontaneous focusing, and then found that doing so led to better competence in cardinality tasks (Hannula, 2005). Increasing spontaneous focusing on numerosity is an example of helping children mathematize their environment (seek out and use the mathemati- cal information in it). Such tendencies can stimulate children’s self-initiated practice in numerical skills because they notice those features and are in- terested in them. Number Word List A common activity in many families and early childhood settings is helping a child learn the list of number words. Children initially may say numbers in the number word list in any order, but rapidly the errors take on a typical form. Children typically say the first part of the list correctly, and then may omit some numbers in the next portion of the list, or they say a lot of numbers out of order, often repeating them (e.g., one, two, three, four, five, eight, nine, four, five, two, six) (Fuson, 1988; Fuson, Richards, and Briars, 1982; Miller and Stigler, 1987; Siegler and Robinson, 1982). Children need to continue to hear a correct number list to begin to include the missing numbers and to extend the list. Children can learn and practice the number word list by hearing and saying it without doing anything else, or it can be heard or said in coordina- tion with another activity. Saying it alone allows the child to concentrate on the words, and later on the patterns in the words. However, it is also helpful to practice in other ways to link the number words to other aspects of the number core. Saying the words with actions (e.g., jumping, pointing, shak- ing a finger) can add interest and facilitate the 1-to-1 correspondences in counting objects. Raising a finger with each new word can help in learning how many fingers make certain numbers, and flashing ten fingers at each decade word can help to emphasize these words as made from tens. Counting: 1-to-1 Correspondences In order to count a group of objects the person counting must use some kind of action that matches each word to an object. This often involves moving, touching or pointing to each object as each word is said. This counting action requires two kinds of correct matches (1-to-1 correspon- dences): (1) the matching in a moment of time when the action occurs and a

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15 PATHS FOR NUMBER, RELATIONS, AND OPERATIONS word is said, and (2) the matching in space where the counting action points to an object once and only once. Children initially make errors in both of these kinds of correspondences (e.g., Fuson, 1988; Miller et al., 1995). They may violate the matching in time by pointing and not saying a word or by pointing and saying two or more words. They may also violate the match- ing in space by pointing at the same object more than once or skipping an object; these errors are often more frequent than the errors in time. Four factors strongly affect counting correspondence accuracy: (1) amount of counting experience (more experience leads to fewer errors), (2) size of set (children become accurate on small sets first), (3) arrange- ment of objects (objects in a line make it easier to keep track of what has been counted and what has not), and (4) effort (see research reviewed in Clements and Sarama, 2007, and in Fuson, 1988). Small sets (initially up to three and later also four and five) can be counted in any arrangement, but larger sets are easier to count when they are arranged in a line. Children ages 2 and 3 who have been given opportunities to learn to count objects accurately can count objects in any arrangement up to 5 and count objects in linear arrangements up to 10 or more (Clements and Sarama, 2007; Fuson, 1988). In groundbreaking research, Gelman and Gallistel (1978) identified five counting principles that stimulated a great deal of research about aspects of counting. Her three how-to-count principles are the three mathematical aspects we have just discussed: (1) the stable order principle says that the number word list must be used in its usual order, (2) the one-one principle says that each item in a set must be tagged by a unique count word, and (3) the cardinality principle says that the last number word in the count list represents the number of objects in the set. Her two what-to-count prin- ciples are mathematical aspects we have also discussed: (1) the abstraction principle states that any combination of discrete entities can be counted (e.g., heterogeneous versus homogeneous sets, abstract entities, such as the number of days in a week) and (2) the order irreleance principle states that a set can be counted in any order and yield the same cardinal number (e.g., counting from right to left versus left to right). Gelman took a strong position that children understand these count- ing principles very early in counting and use them in guiding their count- ing activity. Others have argued that at least some of these principles are understood only after accurate counting is in place (e.g., Briars and Siegler, 1984). Still others, taking a middle ground between the “principles before” view and the “principles after” view, suggest that there is a mutual (e.g., iterative) relation between understanding the count principles and count- ing skill (e.g., Baroody, 1992; Baroody and Ginsburg, 1986; Fuson, 1988; Miller, 1992; Rittle-Johnson and Siegler, 1998). Each of these aspects of counting is complex and does not necessarily exist as a single principle that is understood at all levels of complexity at

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16 MATHEMATICS LEARNING IN EARLY CHILDHOOD once. Children may initially produce the first several number words and not even separate them into distinct words (Fuson, Richards, and Briars, 1982). They may think that they need to say the number word list in order as they count, but early on they cannot realize the implication that they need a unique last counted word, or they would not repeat words so frequently as they say the number word list. The what-to-count principles also cover a range of different under- standings. It takes some time for children to learn to count parts of a thing (Shipley and Shepperson, 1990; Sophian and Kailihiwa, 1998), a later use of the abstraction principle. And the order irrelevance principle (counting in any order will give the same result) seems to be subject to expectations about what is conventional “acceptable” counting (e.g., starting at one end of a row rather than in the middle) as well as involving, later on, a deeper understanding of what is really involved in 1-to-1 correspondence: Count- BOX 5-5 Common Counting Errors There are some common counting errors made by young children as they learn the various principles that underpin successful counting. Counting requires effort and continued attention, and it is normal for 4-year-olds to make some errors and for 5-year-olds to make occasional errors, especially on larger sets (of 15 or more for 4-year-olds and of 25 or more for 5-year-olds). Younger children may initially make quite a few errors. It is much more important for children to be enthusiastic counters who enjoy counting than for them to worry so much about errors that they are reluctant to count. If one looks at the proportion of objects that receive one word and one point, children’s counting often is pretty accurate. Letting errors go sometimes or even somewhat frequently if children are trying hard and just mak- ing the top four kinds of errors is fine as long as children understand that correct counting requires one point and one word for each object and are trying to do that. As with many physical activities, counting will improve with practice and does not need to be perfect each time. Teachers do not have to monitor children’s counting all of the time. It is much more important for all children to get frequent counting practice and watch and help each other, with occasional help and corrections from the teacher. Very young children counting small rows with high effort make more errors in which their say-point actions do not correspond than errors in the matching of the points and objects. Thus, they may need more practice coordinating their actions of saying one word and pointing at an object. Energetic collective practice in which children rhythmically say the number word list and move down their hand with a finger pointed as each word is said can be helpful. To vary the practice, the words can sometimes be said loudly and sometimes softly, but always with emphasis (a regular beat). The points can involve a large motion of the whole arm or a smaller motion, but, again, in a regular beat with each word. Coordinating these actions

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1 PATHS FOR NUMBER, RELATIONS, AND OPERATIONS ing is correct if and only if each object receives one number word (LeFevre et al., 2006). An aspect of the 1-to-1 principle that is difficult even for high school students or adults to execute is remembering exactly which objects they have already counted with a large fixed set of objects scattered irregu- larly around (such as in a picture) (Fuson, 1988). The principles are useful in understanding children’s learning to count, but they should not be taken as simplistic statements that describe knowl- edge that is all-or-nothing or that has a simple relationship to counting skill. It can be helpful for teachers or parents to make statements of vari- ous aspects of counting (e.g., Remember that each object needs one point and one number word, You can’t skip any, Remember where you started in the circle so you stop just before that.). But children will continue to make counting errors even when they understand the task, because counting is a complex activity (see Box 5-5). of saying and pointing is the goal for overcoming this type of error. For variety, these activities can involve other movements, such as marching around the room with rhythmic arm motions or stamping a foot saying a count word each time. Counting an object twice or skipping over an object are errors made occasion- ally by 4-year-olds and even by 5-year-olds on larger sets. These seem to stem from momentary lack of attention rather than lack of coordination. Trying hard or counting slowly can reduce these errors. However, when two counts of the same set disagree, many children of this age think that their second count is correct, and they do not count again. Learning the strategy of counting a third time can increase the accuracy of their counts. If children are skipping over many objects, they need to be asked to count carefully and don’t skip any. Young children sometimes make multiple count errors on the last object. They either find it difficult to stop or think they need to say a certain number of words when counting and just keep on counting so they say that many. When they say the number word list, more words are better, so they need to learn that saying the number word list when counting objects is controlled by the number of objects. Reminding them that even the last object only gets one word and one point can help. They also may need the physical support of holding their hand as they reach to point to the last object so that the hand can be stopped from extra points and the last word is said loudly and stretched out (e.g., fii-i-i-ve) to inhibit saying the next word. Regularity and rhythmicity are important aspects of counting. Activities that increase these aspects can be helpful to children making lots of correspondence errors. Children who are not discouraged about their counting competence gener- ally enjoy counting all sorts of things and will do so if there are objects they can count at home or in a care or education center. Counting in pairs to check each other find and correct errors is often fun for the pairs. Counting in other activities, such as building towers with blocks, should also be encouraged.

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16 MATHEMATICS LEARNING IN EARLY CHILDHOOD siiiixxx 789 To count on like this, a child must shift from the cardinal meaning above to the count meaning of six and then keep counting 7, 8, 9. Trying this with different problems enables many children to see this general pattern and begin counting on. Transition strategies, such as count- ing 1, 2, 3, 4, 5, 6 very quickly or very softly or holding the 6 (siiiiiiixxxxxx), have been observed in students who are learning counting on by themselves; these can be very useful in facilitating this transition to counting on (e.g., Fuson, 1982; Fuson and Secada, 1986; Secada, Fuson, and Hall, 1983). Some weaker students may need explicit encouragement to trust the six and to let go of the initial counting of the first addend, and they may need to use these transitional methods for a while. Counting on has two parts, one for each addend. The truncation of the final counting all by starting with the cardinal number of the first addend was discussed above. Counting on also requires keeping track of the second addend—of how many you count on so that you count on from the first ad- dend exactly the number of the second addend. When the number is small, such as for 6 + 3, most children use perceptual subitizing to keep track of the 3 counted on. This keeping track might be visual and involve actual objects, fingers, or drawn circles. But it can also use a mental visual image (some children say they see 3 things in their head and count them). Some children use auditory subitizing (they say they hear 7, 8, 9 as three words). For larger second addends, children use objects, fingers, or conceptual subi - tizing to keep track as they count on. For 8 + 6, they might think of 6 as 3 and 3 and count with groups of three: 8 9 10 11 12 13 14 with a pause after the 9 10 11 to mark the first three words counted on. Other children might use a visual (I saw  circles and another  circles) or an auditory rhythm to keep track of how many words they counted on. So here we see how the perceptual subitizing and the conceptual subitizing, which begin very early, come to be used in a more complex and advanced mathematical process. This is how numerical ideas build, integrating the levels of thinking visually/holistically and thinking about parts into a complex new concep- tual structure that relates the parts and the whole. Children can discuss the various methods of keeping track, and they can be helped to use one that will work for them. Almost all children can learn to use fingers successfully to keep track of the second addend. Many experiences with composing/decomposing (finding partners hid- ing inside a number) can give children the understanding that a total is any number that has partners (addends) that compose it. When subtracting, they have been seeing that they take away one of those addends, leav- ing the other one. These combine into the understanding that subtracting means finding the unknown addend. Therefore, children can always solve

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165 PATHS FOR NUMBER, RELATIONS, AND OPERATIONS subtraction problems by a forward method that finds the unknown addend, thus avoiding the difficult and error-prone counting down methods (e.g., Baroody, 1984; Fuson, 1984, 1986b). So 14 − 8 = ? can be solved as 8 + ? = 14, and students can just count on from 8 up to 14 to find that 8 plus 6 more is 14. Some first graders will also move on to Level 3 derived fact solution methods (see Box 5-11) such as doubles plus or minus one and the gen- eral method that works for all teen totals: the make-a-ten methods taught in East Asia (see Chapter 4 and, e.g., Geary et al., 1993; Murata, 2004). These make-a-ten methods are particularly useful in multidigit addition and subtraction, in which one decomposes a teen number into a ten to give to the next column while the leftover ones remain in their column. More children will be able to learn make-a-ten methods if they have learned the prerequisites for them in kindergarten or even in Grade 1. The comparison situations compare a large quantity to a smaller quan- tity to find the difference. These are complex situations that are usually not solvable until Grade 1. The third quantity, the difference, is not physically present in the situation, and children must come to see the differences as the extra leftovers in the bigger quantity or the amount the smaller quantity needs to gain in order to be the same as the bigger quantity. The language involved in comparison situations is challenging, because English gives two kinds of information in the same sentence. Consider, for example, the sen- tence Emily has fie more than Tommy. This says both that Emily has more than Tommy and that she has five more. Many children do not initially hear the five. They will need help and practice identifying and using the two kinds of information in this kind of sentence (see the research reviewed in Clements and Sarama, 2007, 2008; Fuson, 1992a, 1992b; Fuson, Carroll, and Landis, 1996). Learning to mathematize and model addition and subtraction situa- tions with objects, fingers, and drawings is the foundation for algebraic problem solving. More difficult versions of the problem situations can be given from Grade 1 on. For example, the start or change number can be the unknown in change plus problems: Joey drew 5 houses and then he drew some more. Now he has  houses. How many more houses did he draw? Children naturally model the situation and then reflect on their model (with objects, fingers, or a drawing) to solve it (see research summarized in Clements and Sarama, 2007; Fuson, 1992a, 1992b). From Grade 2 on they can also learn to represent the situation with a situation equation (e.g., 5 + ? = 9 as in the example above, or ? + 4 for an unknown start number) and then reflect on that to solve it. This process of mathematizing (including representing the situation) and then solving the situation representation is algebraic problem solving.

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166 MATHEMATICS LEARNING IN EARLY CHILDHOOD Issues in Learning Relations and Operations The Extensie Learning Path for Addition and Subtraction The teaching-learning path we describe shows that even the most ad- vanced solution strategies for adding and subtracting single-digit numbers have their roots before age 2 and may not culminate until Grade 1 or even Grade 2. The paths also illustrate how children coordinate several differ- ent complex kinds of understandings and skills beginning with perceptual subitizing through conceptual subitizing and then counting and matching to employ more sophisticated problem-solving strategies. This makes it clear that one cannot characterize the learning of single-digit addition and subtraction as simply “memorizing the facts” or “recalling the facts,” as if children had been looking at an addition table of numbers and memoriz- ing these. Children do remember particular additions and subtractions as early as age 2, but each of these has some history as perceptually or con- ceptually subitized situations, counted situations over many examples, or additions/subtractions derived from other known additions/subtractions. It is therefore much more appropriate to set learning goals that use the ter- minology fluency with single-digit additions and their related subtractions rather than the terms recalled or memorized facts. The latter terms imply simplistic rote teaching/learning methods that are far from what is needed for deep and flexible learning. The Mental Number Word List as a Representational Tool We have demonstrated how children come to use the number word list (the number word sequence) as a mental tool for solving addition and subtraction problems. They are able to use increasingly abbreviated and abstract solution methods, such as counting on and the make-a-ten meth- ods. The number words themselves have become unitized mental objects to be added, subtracted, and ordered as their originally separate sequence, counting, and cardinal meanings become related and finally integrated over several years into a truly numerical mental number word sequence. Each number can be seen as embedded within each successive number and as seriated: related to the numbers before and after it by a linear ordering created by the order relation less than applied to each pair of numbers (see Box 5-12). This is what Piaget (1941/1965) called truly operational cardinal number: Any number in the sequence displays both class inclusion (the embeddedness) and seriation (see also Kamii, 1985). But this fully Piagetian integrated sequence will not be finished for most children until Grade 1 or Grade 2, when they can do at least some of the Step 3 derived fact solution methods, which depend on the whole teaching-learning path we have discussed.

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16 PATHS FOR NUMBER, RELATIONS, AND OPERATIONS BOX 5-12 Ordering and Ordinal Numbers There is frequent confusion in the research literature in the use of the terms ordered or ordering, ordinal number, and order relation. Some of this confusion stems from the fact that adults can flexibly and fluently use the counting, cardinal, and ordinal meaning of number words without needing to consciously think about the different meanings. As a result, they may not be able to differentiate the mean- ings very clearly. But young children learn the meanings separately and need to connect them. When counting to find the total number in a set, the order for connecting each number word to objects is arbitrary and could be done in any order. As noted previously, the last number takes on a cardinal meaning and refers to the total numbers of items counted. Thus, the cardinal meaning of a number refers to a set with that many objects. Cardinal numbers can be used to create an order rela- tion. That is the idea that one set has more members than another set. An order relation (one number or set is less than or more than another number or set) tells how two quantities are related. This order relation produces a linear ordering on these numbers or sets. An ordinal number tells where in the ordering a particular number or set falls. A child can subitize for the small ordinal numbers (see whether an object in an ordered set is first, second, or third), but needs to count for larger ordinal numbers and shift from a count meaning to an ordinal meaning (e.g., count one, two, three, four, five, six, seven [count meaning]. That person is seventh [ordinal meaning and ordinal work] in the line to buy tickets.). We have not emphasized ordinal words in this chapter because they are so much more difficult than are cardinal words, and children learn them much later (e.g., Fuson, 1988). Although 4- and 5-year-olds could learn to use the ordinal words first, second, and last, it is not crucial that they do so. The ordinal words first through tenth could wait until Grade 1. Many researchers have noted how the number word list turns into a mental representational tool for adding and subtracting. A few researchers have called this a mental number line. However, for young children this is a misnomer, because children in kindergarten and Grade 1 are using the number word list (sequence) as a count model: Each number word is taken as a unit to be counted, matched, added, or subtracted. In contrast, a num- ber line is a length model, like a ruler or a bar graph, in which numbers are represented by the length from zero along a line segmented into equal lengths. Young children have difficulties with the number line representa- tion because they have difficulty seeing the units—they need to see things, so they focus on the numbers instead of on the lengths. So they may count the starting point 0 and then be off by one, or they focus on the spaces and are confused by the location of the numbers at the end of the spaces. The report Adding It Up: Helping Children Learn Mathematics (National

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16 MATHEMATICS LEARNING IN EARLY CHILDHOOD Research Council, 2001a) recognized the difficulties of the number line representation for young children and recommended that its use begin at Grade 2 and not earlier. The number line is particularly important when one wants to show parts of one whole, such as one-half. In early childhood materials, the term number line or mental number line often really means a number path, such as in the common early childhood games in which numbers are put on squares and children move along a numbered path. Such number paths are count models—each square is an object that can be counted—so these are appropriate for children from age 2 through Grade 1. Some research sum- marized in Chapter 3 did focus on children’s and adult’s use of the analog magnitude system to estimate large quantities or to say where specified larger numbers fell along a number line. Again, it is not clear, especially for children, whether they are using a mental number list or a number line; the crucial research issue is the change in the spacing of the numbers with age, and this could come either from children’s use of a mental number list or a number line. The use of number lines, such as in a ruler or a bar graph scale, is an important part of measurement and is discussed in Chapter 6. But for numbers, relations, and operations, physical and mental number word lists are the appropriate model. Variability in Children’s Solution Methods The focus of this chapter is on how children follow a learning path from age 2 to Grade 1 in learning important aspects of numbers, relations, and operations. We continually emphasize that there is variability within each age group in the numbers and concepts with which a given child can work. As summarized in Chapter 3, much of this variability stems from differences in opportunities to learn and to practice these competencies, and we stress how important it is to provide such opportunities to learn for all children. We close with a reminder that there is also variability within a given individual at a given time in the strategies the child will use for a given kind of task. Researchers through the years have shown that children’s strategy use is marked by variability both within and across children (e.g., Siegler, 1988; Siegler and Jenkins, 1989; Siegler and Shrager, 1984). Even on the same problem, a child might use one strategy at one point in the session, and another strategy at another point. As children gain proficiency, they gradually move to more mature and efficient strategies, rather than doing so all at once. The variability itself is thought to be an important engine of cognitive change. Similarly, as discussed above, accuracy can vary with effort, particularly with counting. The variability in the use of strategies within or across children can provide important opportunities to discuss different methods and extend understandings of all participants. The vari-

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16 PATHS FOR NUMBER, RELATIONS, AND OPERATIONS ability in results with different levels of effort can lead to discussions about how learning mathematics depends on effort and practice and that everyone can get better at it if they practice and try hard. Effort creates competen- cies that are the building blocks for the next steps in the learning path for numbers, relations, and operations. SUMMARY The teaching-learning path described in this chapter shows how young children learn, integrate, and extend their knowledge about cardinality, the number word list, 1-to-1 counting correspondences, and written number symbols in successive steps from age 2 to 7. Much of this knowledge re- quires specific cultural knowledge—for example, the number word list in English, counting, matching, vocabulary about relations and operations. Children require extensive, repeated experiences with small numbers and then similar experiences with larger and larger numbers. Counting must become very fluent, so that it can become a mental representational tool for problem solving. As we have shown, even young children can have experi- ences in the teaching-learning path that support later algebraic learning. To move through the steps in the teaching-learning path, children require teaching and interaction in the context of explicit, real-world problems with feedback and opportunities for reflection provided. They also require accessible situations in which they can practice (consolidate), deepen, and extend their learning and their own. REFERENCES AND BIBLIOGRAPHY Baroody, A.J. (1984). Children’s difficulties in subtraction: Some causes and questions. Journal for Research in Mathematics Education, 15(3), 203-213. Baroody, A.J. (1987). Children’s Mathematical Thinking: A Deelopmental Framework for Pre- school, Primary, and Special Education Teachers. New York: Teachers College Press. Baroody, A.J. (1992). The development of preschoolers’ counting skills and principles. In J. Bideaud, C. Meljac, and J.P. Fischer (Eds.), Pathways to Number (pp. 99-126). Hillsdale, NJ: Erlbaum. Baroody, A.J., and Coslick, R.T. (1998). Fostering Children’s Mathematical Power: An Ines- tigatie Approach to K- Mathematics Instruction. Mahwah, NJ: Erlbaum. Baroody, A.J., and Gannon, K.E. (1984). The development of the commutativity principle and economical addition strategies. Cognition and Instruction, 1, 321-329. Baroody, A.J., and Ginsburg, H.P. (1986). The relationship between initial meaning and mechanical knowledge of arithmetic. In J. Hiebert (Ed.), Conceptual and Procedural Knowledge: The Case of Mathematics. Hillsdale, NJ: Erlbaum. Baroody, A.J., and Kaufman, L.C. (1993). The case of Lee: Assessing and remedying a numerical-writing difficulty. Teaching Exceptional Children, 25(3), 14-16. Baroody, A.J., Wilkins, J.L.M., and Tiilikainen, S.H. (2003). The development of children’s understanding of additive commutativity: From protoquantitative concept to general con- cept? In A.J. Baroody and A. Dowker (Eds.), The Deelopment of Arithmetic Concepts and Skills: Constructing Adaptie Expertise (pp. 127-160). Mahwah, NJ: Erlbaum.

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10 MATHEMATICS LEARNING IN EARLY CHILDHOOD Baroody, A., Lai, M-L., and Mix, K.S. (2006). The development of young children’s early num- ber and operation sense and its implications for early childhood education. In B. Spodek and O. Saracho (Eds.), Handbook of Research on the Education of Young Children (pp. 187-221). Mahwah, NJ: Erlbaum. Benson, A.P., and Baroody, A.J. (2002). The Case of Blake: Number-Word and Number De- elopment. Paper presented at the annual meeting of the American Educational Research Association, April, New Orleans, LA. Briars, D., and Siegler, R.S. (1984). A featural analysis of preschoolers’ counting knowledge. Deelopmental Psychology, 20(4), 607-618. Carpenter, T.P., Ansell, E., Franke, M.L., Fennema, E.H., and Weisbeck, L. (1993). Models of problem solving: A study of kindergarten children’s problem-solving processes. Journal for Research in Mathematics Education, 2, 428-441. Case, R. (1991). The Mind’s Staircase: Exploring the Conceptual Underpinnings of Children’s Thought and Knowledge. Hillsdale, NJ: Erlbaum. Clements, D.H. (1999). Subitizing: What is it? Why teach it? Teaching Children Mathematics, 5, 400-405. Clements, D.H., and Sarama, J. (2007). Early childhood mathematics learning. In F.K. Lester, Jr. (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 461-555). New York: Information Age. Clements, D.H., and Sarama, J. (2008). Experimental evaluation of a research-based preschool mathematics curriculum. American Educational Research Journal, 5, 443-494. Clements, D.H., Sarama, J., and DiBiase, A. (2004). Engaging Young Children in Mathemat- ics: Findings of the 2000 National Conference on Standards for Preschool and Kinder- garten Mathematics Education. Mahwah, NJ: Erlbaum. DeCorte, E., and Verschaffel, L. (1985). Beginning first graders’ initial representation of arith- metic word problems. Journal of Mathematical Behaior, 1, 3-21. Duncan, A., Lee, H., and Fuson, K.C. (2000). Pathways to early number concepts: Use of 5- and 10-structured representations in Japan, Taiwan, and the United States. In M.L. Fernandez (Ed.), Proceedings of the Twenty-Second Annual Meeting of the North Ameri- can Chapter of the International Group for the Psychology of Mathematics Education, Vol. 2 (p. 452). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education. Fuson, K.C. (1982). An analysis of the counting-on solution procedure in addition. In T. Romberg, T. Carpenter, and J. Moser (Eds.), Addition and Subtraction: A Deelopmental Perspectie (pp. 67-81). Hillsdale, NJ: Erlbaum. Fuson, K.C. (1984). More complexities in subtraction. Journal for Research in Mathematics Education, 15, 214-225. Fuson, K.C. (1986a). Roles of representation and verbalization in the teaching of multi-digit addition and subtraction. European Journal of Psychology of Education, 1, 35-56. Fuson, K.C. (1986b). Teaching children to subtract by counting up. Journal for Research in Mathematics Education, 1, 172-189. Fuson, K.C. (1988). Children’s Counting and Concept of Number. New York: Springer- Verlag. Fuson, K.C. (1992a). Research on learning and teaching addition and subtraction of whole numbers. In G. Leinhardt, R.T. Putnam, and R.A. Hattrup (Eds.) The Analysis of Arith- metic for Mathematics Teaching (pp. 53-187). Hillsdale, NJ: Erlbaum. Fuson, K.C. (1992b). Research on whole number addition and subtraction. In D. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 243-275). New York: Macmillan.

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