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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, childrenâs 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 neededâtime at home or in an early childhood learning centerâthat 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 childrenâs learning will be advanced while othersâ 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 127

128 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 c Â ompose/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.

PATHS FOR NUMBER, RELATIONS, AND OPERATIONS 129 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: Childrenâs 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 childrenâs 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.

130 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 â¤ 5. Step 2 (age 4/prekindergarten): Use counting and matching strategies to find which is more (less) for two numbers â¤ 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 â¤ 10. Step 4 (Grade 1): Solve comparison word problems that ask, âHow many more (less) is one group than another?â for two numbers â¤ 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 â¤ 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 â¤ 8. Step 3 (age 5/kindergarten): Use cardinal counting to solve situation, word, oral number word, and written numeral problems with totals â¤ 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 â¤ 18 and find subtraction as an unknown addend.

PATHS FOR NUMBER, RELATIONS, AND OPERATIONS 131 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).

132 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 toddlersâ 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 âtwo-nessâ in any set of two things. They also need to extend their knowledge to larger and larger groupsâfrom one and two to three, four, and five, although these are more difficult to see and label (Baroody, Lai, and Mix, 2006; Ginsburg, 1989). Childrenâs 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 childâs 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 five). 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

PATHS FOR NUMBER, RELATIONS, AND OPERATIONS 133 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

134 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 childrenâs 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 childrenâs 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

PATHS FOR NUMBER, RELATIONS, AND OPERATIONS 135 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 C Â lements 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 irrelevance 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 âprinciples beforeâ view and the âprinciples afterâ 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

136 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 âacceptableâ 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, childrenâs 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 childrenâs 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

PATHS FOR NUMBER, RELATIONS, AND OPERATIONS 137 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 childrenâs 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 canât 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 donât 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.

138 MATHEMATICS LEARNING IN EARLY CHILDHOOD Written Number Symbols Learning to read written number symbols is quite variable and de- pends considerably on the written symbols in childrenâs environment and how often these are pointed out and read with a number word so that they can learn the symbol-word pair (Clements and Sarama, 2007; Mix, H Â uttenlocher, and Levine, 2002). Unlike much of the number core discussed so far, learning these pairs is rote learning with hardly any possibility of finding and using sequential information. Component parts of particular numbers, or an overall impression (e.g., an 8 looks like a snowman) can be identified and discussed using perceptual learning principles (Baroody, 1987; Baroody and Coslick, 1998; Gibson, 1969; Gibson and Levin, 1975). Learning to recognize the numerals is not a hugely difficult task, and 2- and 3-year-olds can often read some numerals; 4-year-olds can learn to read many of the numerals to 10. Kindergarten children with such experiences can then concentrate on reading and understanding the numerals for the teens, and first graders can master the cardinal tens and ones connections in the numerals from 20 to 100 (see discussions at those levels). Learning to write number symbols (numerals) is a much more difficult task than is reading them and often is not begun until kindergarten. Writing numerals requires children to have an accurate mental image of the symbol, which entails left-right orientation, and a motor plan to translate the mental image into the correct sequence of motor actions to form a numeral (e.g., see details in Baroody, 1987; Baroody and Coslick, 1998; Baroody and Kaufman, 1993). Some numerals are much easier than others. The loops in 6 and 9, the curve and straight line in the 2, and the crossovers in the 8 are difficult but can be mastered by kindergarten children with effort. The easier numerals 1, 3, 4, 5, and 7 can often be mastered earlier. Whenever children do learn to write numerals, learning to write correct and readable numerals is not enough. They must become fluent at writing numerals (i.e., writing numerals must become overlearned) so that writing them as part of a more complex task is not so slow or effortful as to be discouraging when solving several problems. It is common for children at this step and even later to reverse some numerals (such as 3) because the left-right orientation is difficult for them. This will become easier with age and experience. Coordinating the Components of the Number Core We discussed above how children coordinate their knowledge of the number word list and 1-to-1 correspondences in time and in space to count groups of objects in space. They also gradually generalize what they can count and extend their accurate counting to larger sets and to sets in vari- ous arrangements not in a row (circular, disorganized). However, accuracy for the latter comes quite late, except for small sets (Fuson, 1988). Gener-

PATHS FOR NUMBER, RELATIONS, AND OPERATIONS 139 alization of counting involves taking as a unit each object they are counting so that each object can receive one count word. For example, when they are counting toy animals, each animal is a unit regardless of how big it is, what color it is, or what kind of animal it is. Later, 2- and 3-year-olds continue to generalize the range of objects they can count. Children with little experi- ence with print may have more difficulty counting pictures of objects rather than objects themselves, and so they may especially need practice counting pictures of objects (Murphy and Wood, 1981). The next crucial coordination of components is connecting counting and cardinality (Fuson, 1988; Gelman and Gallistel, 1978). When counting things (objects or pictures), the counting action matches each count word to one thing (see discussion above and in Chapter 2). But a cardinal num- ber word refers to how many things there are in the whole set of things. So when anyone counts, they must at the end of the counting action make a mental shift from thinking of the last counted word as referring to the last counted thing to thinking of that word as referring to all of the things (the number of things in the whole set, i.e., the cardinality of the set). For example, when counting 7 toy animals 1, 2, 3, 4, 5, 6, 7, the 7 refers to the one last animal you count when you say 7. But then you must shift to think- ing of all of the animals and think of the 7 as meaning all of them: There are 7 animals. This is a major conceptual milestone for young children. When children discover this relationship, they tend to apply it to all counts no matter the size of the set of objects (Fuson, 1988). Therefore, this is a type of rule/principle of learning that children immediately generalize and apply fairly consistently. It is relatively easy to teach children that the last word said in counting tells how many there are (see Fuson, 1988). For example, a statement of this principle followed by three demonstrations followed by another statement of the principle was sufficient to move 20 of 22 children ages 2 years 8 months to 3 years 11 months who did not use the principle to using it (Fuson, 1988). However, not all children really understand cardinality, even when they understand the importance of the last counted word (Fuson, 1988). Some children initially understand only that the last word answers the âHow many?â question. They do not fully grasp the more abstract idea of cardi- nality. Thus, they give their last counted word when asked how many there are, but they do not point to all of the objects when asked the cardinal- ity question âShow me the seven animals.â Instead, they point at the last animal again. It is important to note that responding with the last word is progress. Earlier when asked âHow many are there?â children may have recounted or given a number other than the last counted word. Children who recount are understanding the question âHow many are there?â as a request to count, not as a cardinal request. Such children may recount several if the question is repeated and may protest But I already did it or I already said it because they donât understand the reason for the repeated

140 MATHEMATICS LEARNING IN EARLY CHILDHOOD requests (to them, each count is a correct response to the How many are there? question). Children making the other error (giving a number that dif- fers from the last word) are understanding that the question How many are there? is a request for cardinal information about the whole set, but they do not yet understand that the cardinal information is given by counting, and, in particular, by the last word said in counting. Verbal knowledge is also required for full competence in discriminat- ing the use of individual number words for each thing counted versus the use of the final number word to refer to the whole set. Even children who gesture correctly to show their count meaning (gesture to one thing) or their cardinal meaning (gesture to the whole set) may struggle with correct verbal expressions (see Box 5-6). Mastering these is a later achievement that will be learned with modeling and practice. BOX 5-6 Learning the Correct Counting Language Learning the singular and plural forms that go with counting (single) and with cardinal (plural) references to objects takes some time. Here are typical examples of errors that children initially make while they are sorting out all of these concep- tual and linguistic issues. After children counted a row of objects, they were asked a count-reference question and a cardinality-reference question (the order varied across children). The count-reference question was Is this the soldier (chip) where you said n? where n was the last word said by the child. The experimenter asked the question three times and pointed to the last item, the next-to-last item, and all the items in the row. The cardinal-reference question was Are these the n soldiers (chips)? The correct answer was always in the middle, because research indicated that young children have a strong bias toward choosing the last alternative. In the examples below, children spontaneously verbalized cardinality or counting refer- ences that disagreed with their gesture. Response to cardinality question: hose are five soldiers, said as child points to T the last soldier. Response to cardinality question: his oneâs the five chips, said as child points T to the last chip. Response to cardinality question: his is the six soldiers, said as child points to T each soldier (said six times). Response to cardinality question: his is the four chips, said as child points to T the last chip. Response to cardinality question: his is where I said chip four, said as childâs T hands gesture to all of the chips. Response to count question: All of these animals I said five. SOURCE: Fuson (1988, p. 232).

PATHS FOR NUMBER, RELATIONS, AND OPERATIONS 141 Step 2 (Age 4 or Prekindergarten) As children become acquainted with the components of number, they extend cardinal counting and conceptual subitizing to larger numbers. The major advances for children at this step who have had opportunities at home or in a care center to learn the previous foundational and achievable number core content involve extending their competency to larger numbers. This means that teachers or caregivers who must support children at differ- ent levels, or support a mixture of children who have learned and those who have not had sufficient opportunity to learn the previous number core con- tent, can frequently combine these groups by allowing children to choose set sizes with which they feel comfortable and can succeed (see Box 5-7). BOX 5-7 Step 2 in the Number Core Age 4 or Prekindergarten Extend Cardinal Counting and Conceptual Subitizing to Larger Numbers 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. Cardinality: Extends conceptual subitizing to 5-groups with 1, 2, 3, 4, 5 to see 6 through 10: can see the numbers 6, 7, 8, 9, 10 as 5 + 1, 5 + 2, 5 + 3, 5 + 4, 5 + 5 and can relate these to the fingers (5 on one hand). May do other such numerical compose/decompose patterns also. Number word list: Continues to extend and learns the irregular teen patterns and extends the early decade twenty to twenty-nine, etc., pattern to higher decades: says 1 to 39. 1-to-1 counting correspondences: Continues to generalize to counting new things and to extend accurate correspondences to larger sets (accuracy will vary with effort): counts accurately 1 to 15 things in a row. Written number symbols: Continues to learn new symbols if given such learning opportunities: reads 1 to 10; writes some numerals. Reverses the cardinal counting principle (the count-to-cardinal shift) to count out n things (makes the cardinal-to-count shift): Must have fluent counting to have the attentional space to remember the number to which youâre counting so you can stop there.

142 MATHEMATICS LEARNING IN EARLY CHILDHOOD Cardinality Children at this level continue to extend to larger numbers their con- ceptual subitizing of small groups to make a larger number, for example, I see one thumb and four fingers make my five fingers (this is part of the relation and operation core and is discussed more there). The 5-groups are particularly important and useful. These 5-groups provide a good way to understand the numbers 6, 7, 8, 9, 10 as 5 + 1, 5 + 2, 5 + 3, 5 + 4, 5 + 5 (see Figure 5-1). The convenient relationship to fingers (5 on one hand) provides a kinesthetic component as well as a visual aspect to this knowledge. Without focused experience with 5-groups, childrenâs notions of the numbers 6 through 10 tend to be hazy beyond a general sense that the numbers are getting larger. Knowing the 5-groups is helpful at the next level, as children add and subtract numbers 6 through 10; the patterns are problem-solving tools that can be drawn or used mentally. Children in East Asia learn and use these 5-group patterns throughout their early numerical learning (Duncan, Lee, and Fuson, 2000). Children can continue to experi- ence and begin remembering other addends that make totals (e.g., 3 and 3 make 6, 8 is 4 and 4). Number Word List As noted, beyond the first ten words, which are arbitrary in most languages (e.g., see the extensive review in Menninger, 1958/1969), most languages begin to have patterns that make them easier to learn. English, however, has irregularities that are challenging for children. A major dif- ficulty in understanding the meaning of the teens words is that English words do not explicitly say the ten that is in the teen number (teen does not mean ten even to many adults), so English-speaking children can benefit 5-groups that show 6 as 5 + 1, 7 as 5 + 2, 8 as 5 + 3, 9 as 5 + 4, and 10 as 5 + 5 6 7 8 9 10 ooooo ooooo ooooo ooooo ooooo o oo ooo oooo ooooo FIGURE 5-1â Five groups to understand the numbers 6, 7, 8, 9, and 10.

PATHS FOR NUMBER, RELATIONS, AND OPERATIONS 143 from visual representations that show the ten inside teen numbers in order to understand what quantities these words represent (see the discussion in the kindergarten level). There are two patterns in the English number words from 20 to 100 that children need to understand if each word is to have its value as some number of tens and some number of ones, as in Chinese words (52 is said as five ten two). One is the irregular pattern in the decade words that name the tens multiples: twenty (twin-tens), thirty (three-tens), forty, fifty (five tens), sixty, seventy, eighty, ninety. As with the teens, the relationships of the decade words to the numbers below ten become really clear only for the last four words because only then are the six, seven, eight, nine said. The irregularities in twenty through fifty interfere with seeing the meaning of these words as two tens, three tens, four tens, five tens, etc., and thus with learning these in order by using the list below ten, as Chinese-speaking children can do (see Chapter 4). Also, as with the teen words, the ten is not said explicitly but is said as a different suffix, âty. Therefore, as discussed later for Grade 1, children need to work explicitly with groups of tens and ones to understand these meanings for the number words from 20 to 100. The second pattern is the pattern of a decade word followed by the decade word with the numbers one through nine: twenty, twenty-one, twenty-two, twenty-three, . . . , twenty-nine. Children can begin to learn this second pattern quite early. Because the transition to ten and the teens words is not clear in English, children often initially do not stop at twenty- nine but continue to count twenty-nine, twenty-ten, twenty-eleven, twenty- twelve, twenty-thirteen (Fuson, 1988). This error can be a mixture of not yet understanding that the pattern ends at nine and difficulty stopping the usual counting at nine in order to shift to another decade. Children in the United States tend to learn the pattern of the decade word followed by a number (1-9) before learning the order of the decade words (e.g., Fuson, 1988; Fuson, Richards, and Briars, 1982; Miller and Stigler, 1987; Siegler and Robinson, 1982). Although some 2- and 3-year- olds begin learning and practicing the patterns for the teens and decade words, the teen pattern can be mastered by almost all 4-year-olds with support and practice, as can the early decades (two cycles of the pattern from twenty through thirty-nine). Many 4-year-olds learn more than this, but mastering the correct order of the decades and using this with the n-ty through n-ty-nine pattern is for many children a kindergarten achievement (e.g., Fuson, 1988; Fuson, Richards, and Briars, 1982; Miller et al., 1995). Structured learning experiences can decrease the time it takes to learn this pattern of decades to 100, but without such experiences this learning effort can continue even to age 6. Counting by tens to 100 to learn this decade sequence is a goal for kindergarten and is discussed in that section.

144 MATHEMATICS LEARNING IN EARLY CHILDHOOD Counting Correspondences At this step, children extend considerably the set size they are able to count accurately. They move from considerable inaccuracy with counting larger sets to only occasional errors, even with large sets of 15 and above, unless the sets are arranged in a disorganized way and children are not able to move objects to keep track of which have been counted (i.e., make a counted and an uncounted pile) (Fuson, 1988). As before, effort continues to be important. Children who are tired or discouraged may make many more errors than they make after a simple prompt to try hard or count slowly. Children at this step also continue to generalize what they can count. Children at this step are working on counting linear arrangements cor- rectly in the teens or above, and many make few errors, showing consider- ably more accuracy than children a year younger (Clements and Sarama, 2007; Fuson, 1988). Of course, accurate counting also depends on knowing an accurate number word list, so accuracy with these larger sets depends on three things: 1. Knowing the patterns discussed above in the number word list so that a correct number word list can be said. 2. Correctly assigning one number word to one object (1-to-1 correspondence). 3. Keeping track of which objects have already been counted so that they are not counted more than once. Differentiating counted from uncounted entities is most easily done by moving objects into a counted set, but this is not possible with things that cannot be moved, such as pictures in a book. For pictures or objects that cannot be moved, counting objects arranged in a row is easiest because one can start at the end of a row and continue to the other end. However, if objects are arranged in a circle, children may initially count on and on around the circle. Strategies for keeping track of messy, large sets continue to develop for many years (Fuson, 1988), with even adults not being en- tirely accurate. Children in kindergarten who have had adequate counting experiences earlier continue to extend their counting of objects as high as 100, often with correct correspondences (and perhaps occasional errors). There may or may not still be errors in the number word list. Written Number Symbols Children at this step continue to extend the number of written number symbols they can read, now often reading many of the numerals 1, 2, 3,

PATHS FOR NUMBER, RELATIONS, AND OPERATIONS 145 4, 5, 6, 7, 8, 9, and 10. However, the 10 at this level means ten ones, the counted number ten that comes after nine. Not until the next level does it come to mean what the 10 symbols actually say: 1 ten and 0 ones. Children at this level can begin to write some numerals, often beginning with the easier numerals 1, 3, 4, and 7. Counting Out ânâ Things Children at this level make one major conceptual advance. They move from knowing that the last number stated represents the amount in the group to knowing how to count out a given number of objects (Clements and Sarama, 2007; Fuson, 1988). Lots of counting of objects and saying the number word list enables their counting to become fluent enough that they can count out a specified number of things, for example, count out 6 things. Counting out n things requires a child to remember the number n while counting. This is more difficult for larger numbers because the child has to remember the number longer. So children may initially count past n because their counting is not fluent (overlearned) enough to count a long sequence of words, remember a number, and monitor with each count whether they have reached the number yet. Counting out a specified number is needed for solving addition and subtraction problems and for doing various real-life tasks, so this is an important milestone. Children can practice this concep- tual task by counting out n things for various family and school purposes; such practice can also occur in game-like activities. Counting out n things also requires a conceptual advance that is the reverse of learning that the last count word tells how many there are. To count out 6 things, a child is being told how many there are (a cardinal meaning) and must then shift to a count meaning of that 6 in order to moni- tor the count words as they are said (Have I said 6 yet?) so that they can stop when they say 6 as a counting word that corresponds to one object. They then have the set of 6 things they need. Step 3 (Kindergarten) At this step children work to integrate all of the core components of number. They are able to see that teen numbers are made up of tens and some ones. They also can come to understand that ten ones make one group of ten (see Box 5-8). Kindergarten children can begin the process with seeing and making tens in teen numbers, and first graders can continue the process for tens and ones in numbers 20 to 100. At both grades this process helps children integrate the number components into a related web of cardinal, counting, and written number symbol knowledge. The first conceptual step is for chil-

146 MATHEMATICS LEARNING IN EARLY CHILDHOOD BOX 5-8 Step 3 in the Number Core Age 5 or Kindergarten Integrate All Core Components, See a Ten and Some Ones in Teen Numbers, Relate Ten Ones to One Ten, and Extend the Core Components to Larger Numbers 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. Cardinality: Extends conceptual subitizing to a new visual group, a group of tens: can see a ten in each teen number (18 = 10 + 8). Number word list: Extends to learn all of the decades in order as a new number word list counting by tens; uses this decade order with the decade pattern to count to 100 by ones: says the tens list 10, 20, 30, . . . , 90, 100; says 1 to 100 by ones. 1-to-1 counting correspondences: Continues to extend accurate correspon- dences to larger sets; accuracy will still vary with effort: counts 25 things in a row with effort. Written number symbols: Coordinates knowledge of symbols 1 to 9 to write teen numbers: reads and writes 1 to 19; reads 1 to 100 arranged in groups of ten when counting 1 to 100. Integrates all of the above for teen numbers so that ten ones = 1 ten, relating the unitary cardinality relationship ten ones + eight ones make eighteen ones to the written symbols 18 as 10 with an 8 on top of the 0 ones in ten. dren to understand each cardinal teen number as consisting of two groups: 1 group of ten things and a group of the ones (the extra over ten). So, for example, 11 is 1 group of ten and 1 one (11 = 10 + 1), 14 is 1 group of ten and 4 ones (14 = 10 + 4), and 18 is 1 group of ten and 8 ones (18 = 10 + 8). The second crucial understanding that builds on the above is that ten ones equal one ten. That is, the written teen number symbols such as 18 mean 1 group of ten (1 ten rather than ten ones) and 8 ones. Being able to see ten ones as one ten is a crucial step on the learning path. It can be helpful for English-speaking children to have experiences seeing 18 things separated into ten and eight and relating these quantities to both the number words âeighteen is ten and eightâ and to the written number symbols (18). It may also be helpful to use the written symbol ver- sion of this as 18 = 10 + 8. Repeated experiences with all of these relation-

PATHS FOR NUMBER, RELATIONS, AND OPERATIONS 147 ships can help children overcome the second kind of typical error in writing teen numbers, in which children write first what they say first. They hear eighteen and know that teens have a 1 in them (they may not yet think of this as one ten) and so they write 81. Kindergarten children can also experience and learn all of the decade words in order from 20 to 100. Doing so while looking at a list of these number symbols grouped in tens can help to reinforce the pattern of the groups of ten. Many states require that kindergarten children understand some as- pects of money, but sometimes they have goals that are not sensible for this age group, even children who have had strong earlier mathematical experiences. The mathematical aspects of money that are most appropriate are the groups of ten pennies in dimes and the groups of five pennies in nickels. Children have been working with these cardinal groups of tens in this level and with 5-groups in the 4-year-old/prekindergarten level, so it is easy to build this understanding by extending this knowledge to coins by using any visual support that relates a 5-group of pennies to one nickel and one 10-group of pennies to one dime. Such supports were used successfully for first graders to construct the relationships for understanding two-digit numbers described next for first graders (Fuson, Smith, and Lo Cicero, 1997; Hiebert et al., 1997). Learning the values of a dime and a nickel are of course particularly complicated because their values are not in the order of the sizes of the coins. In size, a dime < a penny < a nickel, but in value a penny < a nickel < a dime. For this reason, it is too difficult to work with these coins alone rather than with visual supports that show the values of these coins in pen- nies, as discussed above. Counting mixed collections of dimes, nickels, and pennies requires shifting counts from counting by tens when counting dimes to counting by fives when counting nickels to counting by ones when count- ing pennies. Such shifts are too complex for many children at this level, es- pecially if they are looking at the coins rather than looking at their values as pennies. Practice just on the names of the coins and on their visual features, rather than on their value as ones, fives, or tens, is also not appropriate. It is the quantitative values that are mathematically important. Step 4 (Grade 1) At this step children see, say, count, and write tens and ones from 1 to 100 (see Box 5-9). To do this, they build on the integrations among cardinality, counting, and written number symbols that they have made in kindergarten. The major advance has two parts. First, children learn to count by two different units, units of ten and units of one. Second, they learn to shift from counting by units of ten to counting by units of one so that they can count cardinal sets up to 100. Children who have mastered

148 MATHEMATICS LEARNING IN EARLY CHILDHOOD BOX 5-9 Step 4 in the Number Core Grade 1 See, Say, Count, and Write Tens-Units and Ones-Units from 1 to 100 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. Cardinality: Relates patterns in number word list to 100 to quantities of tens and of ones: can see the tens and ones quantities in numbers from 10 to 99 (e.g., 68 = 60 + 8); sees the 60 both as 60 ones (sixty) and as 6 tens; can make drawn quantities to show tens and ones. Number word list: May count groups of ten using a tens list (1 ten, 2 tens, etc.) as well as the decade list 10, 20, 30, . . . . 1-to-1 counting correspondences: Extends counting single units to counting a group of ten as a 10-unit and shifts from counting these units of ten to counting by ones when counting left-over ones units: arranges things in groups of ten (or uses prearranged groups or drawings) and counts the groups by tens and then shifts to a count by ones for the leftover single things: 10, 20, 30, 40, 50, 60, 61, 62, 63, 64, 65, 66, 67, 68, or 1 ten, 2 tens, 3 tens, 4 tens, 5 tens, 6 tens, 6 tens and 1 one, 6 tens and 2 ones, 6 tens and 3 ones, 6 tens and 4 ones, 6 tens and 5 ones, 6 tens and 6 ones, 6 tens and 7 ones, 6 tens and 8 ones. Written number symbols: Extends reading and writing to all two-digit numbers 1 to 99 and understands that the tens digit refers to groups of tens and the ones digit refers to groups of ones; also sees that the 0 from the tens number is hiding behind the ones number so can see 68 as 60 + 8. Integrates all of the above for numbers 1 to 100 so that nâty = n tens (e.g., 60 is 6 tens); the counting by tens and by ones represents sets of tens and of ones; a 2-digit numeral like 68 = 60 + 8 and 68 also means 6 tens and 8 ones. the kindergarten concept that ten ones equal one ten can learn to use visual representations of tens that show each ten as one ten. Children at this step need to be able to make drawings of tens and of ones so that they can represent numbers to use when adding and subtract- ing. Making such drawings can also help with the consolidation of the two-digit numerals, for example, 68 = 60 + 8 as sixty plus eight and as six tens plus eight. Place value cards in which the ones card covers the 0 in the tens card can also help eliminate the typical errors of children hearing 68 as sixty eight and therefore writing what they hear: 608 instead of 68.

PATHS FOR NUMBER, RELATIONS, AND OPERATIONS 149 THE RELATIONS AND OPERATIONS CORE The main mathematical categories in the relations and operations core were discussed in Chapter 2, and the steps through which our four age groups move were summarized in Box 5-2. These steps are elaborated in Box 5-10. In the relations core, children learn to perceive, say, discuss, and create the relations more than, less than, and equal to on two sets. Initially they use general perceptual, or length, or density strategies to decide whether one set is more than, less than, or equal to another set. Gradually these are replaced by more accurate strategies: They match the entities in the sets to find out which has leftover entities, or they count both sets and use under- standings of more than/less than order relations on numbers (see research reviewed in Clements and Sarama, 2007; Fuson, 1988). Eventually, in Grade 1, children begin to see the third set potentially present in relational situations, the difference between the smaller and the larger set (see research reviewed in Fuson, 1992a, 1992b). In this way, relational situations become the third kind of addition/subtraction situations: comparison situations. In the operations core, children learn to see addition and subtraction situations in the real world by focusing on the mathematical aspects of those situations and making a model of the situation (called mathematizing these situations, as explained in Chapter 2). Initially such mathematizing may involve only focusing on the number of objects involved rather than on their color or their use (I see two red spoons and one blue spoon) and using those same objects to find the answer by refocusing on the total or counting it (I see three spoons in all). The three types of addition/Âsubtraction situ- ations that children must learn to solve were discussed in Chapter 2 and summarized in Box 2-4. These types are change plus/change minus, put together/take apart (sometimes called combine), and comparisons. Addition and subtraction situations, and the word problems that de- scribe such situations, provide many wonderful opportunities for learning language. Word problems are short and fairly predictable texts, so children can vary words in them while keeping much of the text. This enables them to say word problems in their own words and help everyoneâs understand- ing. English language learners can repeat such texts and vary particular words as they wish, all with the support of visual objects or acted-out situations. Although children need to learn the special mathematics vocabu- lary involved in addition and subtraction, these problems also give them wonderful opportunities to integrate art (drawing pictures) and language practice and pretend play while also generalizing their growing mathemati- cal knowledge.

150 MATHEMATICS LEARNING IN EARLY CHILDHOOD BOX 5-10 Steps in Addition/Subtraction Operations and Relations Step 1 (ages 2 and 3) â¢ Use subitized and counted cardinality to solve situation and oral number word problems with totals â¤ 5. â¢ Act out numerical situations with objects and say them in words; see answer at the end. â¢ Determine that something is bigger or has more using perceptual, length, and density strategies. Examples of problems they can solve: â¢ Change plus: Two blocks and two blocks make four blocks. â¢ Change minus: Four apples take away one apple is three apples. â¢ Put together/take apart: I see three apples. I see two and one make three. Step 2 (age 4/prekindergarten) â¢ Use conceptual subitizing and cardinal counting to solve situation, word, and oral number word problems with totals â¤ 8. â¢ Solve numerical situations and word problems by modeling actions with objects, fingers, or mentally (or just know the answer); or see or count the answer. â¢ Solve number word problems by modeling actions with objects, fingers, or mentally (or just know the answer); or see or count the answer. â¢ Learn the partners for 3, 4, 5 (e.g., 5 = 4 + 1, 5 = 3 + 2). â¢ For relations, understand and say this is/has less/fewer than that. â¢ For more than/less than relations with totals â¤ 5, act out or show situation, and count or match to solve. Examples of problems they can solve: â¢ Change plus: Two and two make ? Levels in Childrenâs Numerical Solution Methods There is a large research base from around the world describing three levels through which childrenâs numerical solution methods for addition and subtraction situations move (e.g., see the research summarized in ÂBaroody, 1987, 2004; Baroody, Lai, and Mix, 2006; Clements and Sarama, 2007, 2008; Fuson, 1988, 1992a, 1992b; Ginsburg, 1983; Saxe, 1982; Sophian, 1984). These levels are summarized in Box 5-11. At all levels, the solution methods require mathematizing the real-world situation (or later the word problem or the problem represented with numbers) to focus on only the

PATHS FOR NUMBER, RELATIONS, AND OPERATIONS 151 â¢ Change minus: Four take away one is ? â¢ Put together/take apart: Three has ? and ? Step 3 (Kindergarten) â¢ Use conceptual subitizing and cardinal counting to solve situation, word, oral number word, and written numeral problems with totals â¤ 10. â¢ For word problems, model action with objects or fingers or a math drawing and count or see to solve; write an expression or equation. â¢ For oral or written numeral problems, use fingers, objects, or a math drawing to solve. â¢ Engage in learning the partners for 6, 7, 8, 9, 10. â¢ For relations, act out or show with objects or a drawing, then count or match to solve. â¢ Use =, â symbols. Step 4 (Grade 1) â¢ Use Level 2 or Level 3 solution procedures: count on or use a derived fact method for problems with totals â¤ 18 and find subtraction as an unknown addend. â¢ Solve change plus problems by counting on to find the total 6 + 3 = ? â¢ Solve change minus problems by counting on to find the unknown addend 9 â 6 = ? is 6 + ? = 9. â¢ Solve put together/take apart problems by counting on to find the unknown addend 6 + ? = 9. â¢ Advanced first graders use Level 3 solution procedures: (a) doubles and doubles Â± 1. (b) they experience make-a-ten methods: 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14; 14 â 8 is 8 + ? = 14, so 8 + 2 + 4 = 14, so ? = 6 (not all children master these in Grade 1). â¢ Solve comparison situations or determine how much/many more/less by counting or matching for totals â¤ 10, then for totals â¤ 18. mathematical aspectsâthe numbers of things and the additive or subtrac- tive operation in the situation. As we discuss each level, we also describe ways in which children can be helped to learn methods appropriate for that level and the prerequisite knowledge. Children need opportunities to relate strategies to actual objects or pictures of objects and to discuss and explain their thinking. The solution methods at Level 1 use direct modeling of every object. In direct modeling children must carry out the actions in the situation using actual objects or fingers. Until around age 6, children primarily use such direct modeling to solve situations presented in objects, word problems

152 MATHEMATICS LEARNING IN EARLY CHILDHOOD BOX 5-11 Levels in Childrenâs Numerical Solution Methods Level 1: Direct modeling of all quantities in a situation; used at the first three number/operation levels: Counting all: Count out things or fingers for one addend, count out things or fingers for the other addend, and then count all of the things or fingers. Take away: Count out things or fingers for the total, take away the known ad- dend number of things or fingers, and then count the things or fingers that are left. Level 2: Count on can be done in first grade (some children can do so earlier): They use embedded number understanding to see the first addend within the total and so see that they do not need to count all of the total, but instead could make a cardinal-to-count shift and count on from the first addend. Count on to find the total: On fingers or with objects or with conceptual subi- tizing, children keep track of how many words to count on so that they stop when they have counted on the second addend number of words and the last word they say is the total: 6 + 3 = ? would be âsix, seven, eight, nine, so the total is nine. I counted on 3 more from 6 to make 9.â After learning counting on from the first addend, children learn to count on from the larger addend. Count on to find the unknown addend: Children stop counting when they say the total, and the fingers (or other keeping track method) tell the answer (the unknown addend number of words they counted on past the first addend). 6 + ? = 9 would be âsix, seven, eight, nine, so I added on 3 to 6 to make 9. I counted on 3 more from 6 to make 9. Three is my unknown addend.â (situations expressed in words, perhaps with an accompanying picture), oral numerical problems such as three plus two, and written numerical problems such as 3 + 2. Chapter 4 summarized research reporting that more children from low-income families had trouble with the last three kinds of problems than with the first kind and than did their middle-income peers. Therefore, such children especially need help and practice in generating models using objects or fingers for such situations. At Grade 1, children who have not yet moved to the Level 2 general counting on methods (see Box 5-8 and Box 5-11 for more details) can do so with help. In these methods, children shift from the cardinal meaning of the first addend to the counting meaning as they count on from it: For 5 + 2, they think five, shift to the counting word five in the number word list, and count on two more wordsâfive, six, seven. This ability to count on can be

PATHS FOR NUMBER, RELATIONS, AND OPERATIONS 153 Level 3: Derived fact methods in which known facts are used to find related facts (mastery by some/many at first grade). Doubles are totals of two of the same addend: 1 + 1, 2 + 2, 3 + 3, etc., up to 9 + 9. These are learned by many children in the United States because of the easy pattern in their totals (2, 4, 6, 8, etc.). Doubles Â± 1 is a Level 3 more advanced strategy that uses a related double to find the total of two addends in which one addend is one more or less than the other addend (6 + 7 = 6 + 6 + 1 = 12 + 1 = 13). Make-a-ten methods are general methods for adding or subtracting to find a teen total by changing a problem into an easier problem involving 10. Chil- dren first make a 10 from the first addend and then learn to make a 10 from the larger addend. Make a ten to find a total: 8 + 6 becomes 10 + 4 by separating the 6 into the amount that makes 10 with the 8. Then solving 6 = 2 + ? gives the leftover 4 within the 6 to become the ones number in the teen total: 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14. Make a ten to find an unknown addend: 14 â 8 = ? is 8 + ? = 14, so 8 + 2 is 10 plus the 4 in 14 makes 14. So 8 + 6 = 14. In this method subtraction requires adding, which is easier than making a ten to find a total. The first step can also be thought of as subtracting the 8 from 10. Three prerequisites for fluency with make-a-ten methods can be built up before first grade: 1. knowing the number that makes 10 (the partner to 10) for each num- ber 3 to 9; 2. knowing each teen number as a 10 and some ones (e.g., knowing that 14 = 10 + 4 and that 10 + 4 = 14 without counting); and 3. knowing all the partners of numbers 3 to 9 so that the second number can be broken into a partner to make 10 and the leftover partner that will make the teen number. facilitated by childrenâs earlier work with embedded number experiences of finding partners of a total (e.g., Inside seven, I see five and two) and by fluency with the count word sequence, so they can begin counting from any number (most 2-, 3-, and 4-year-olds need to start at 1 when counting and cannot start from just any number). With larger second addends, children also need a method of keeping track of how many they have counted on. These counting on methods are sufficient for all further quantitative work, especially if children are helped to see subtraction as finding an unknown addend, so that they can use counting on to find that addend. Counting down to subtract is difficult, and children make many errors at it (Baroody, 1984; Fuson, 1984). Just counting backward is difficult, and children make various count-cardinal errors in counting down. Counting forward to find an unknown addend for subtraction (e.g., solving 9 â 5 = ? as 5 + ? = 9)

154 MATHEMATICS LEARNING IN EARLY CHILDHOOD is much easier and can make subtraction as easy as addition (e.g., Fuson, 1986b; Fuson and Willis, 1988). It also emphasizes addition and subtrac- tion as inverse operations. The derived fact methods (Level 3) are mastered by some children at Grade 1, depending on how many of the prerequisites shown in Box 5-8 have been made accessible for 4- and 5-year-olds and then have been practiced so that they become fluent. These methods require recomposing the given numbers into a new, easier problem (e.g., 9 + 4 becomes 10 + 3). The make-a-ten methods are taught in East Asian countries and are very useful in multidigit computation (see the discussion in Chapter 2). The prerequisites are discussed later in the summaries of the 4- and 5-year-olds because children can begin building these prerequisites then. Enabling 4- and 5-year-olds to learn the prerequisites for the counting on and derived facts methods can help low-income children to learn more advanced strate- gies, which fewer of them do now. This can also help children with learning difficulties in mathematics because they often continue to use the Level 1 modeling methods for too many years unless they are helped to learn more advanced strategies. The general counting on methods for addition and subtraction can be learned meaningfully and done accurately and rapidly by most children in Grade 1 (Fuson, 2004). Throughout the process of learning and using more advanced ap- proaches to solving addition and subtraction problems, children also be- come fluent with individual sums and differences. Small numbers, such as plus 1 and minus one, and doubles (2 + 2, 3 + 3) become fluent early. Others become fluent over time. Step 1 (Ages 2 and 3) Children at this step use subitized and counted cardinality to solve situa- tion and oral number word problems. They also use perceptual, length, and density strategies to find which is more with totals â¤ 5 (see Box 5-10). Relations: More Than, Equal To, Less Than Children ages 2 and 3 begin to learn the language involved in rela- tions (Clements and Sarama, 2007, 2008; Fuson, 1992a, 1992b; Ginsburg, 1977). More is a word learned by many children before they are 2. Initially it is an action directive that means: Give me more of this. But gradually children become able to use perceptual subitizing and length or density strategies to judge which of two sets has more things: She has more than I have. Such comparisons may not be correct at this age level if the sets are larger than three because children focus on length or on density and cannot yet coordinate these dimensions or use the strategies of matching or count-

PATHS FOR NUMBER, RELATIONS, AND OPERATIONS 155 ing effectively (see research reviewed in Fuson, 1988, 1992a, 1992b, and in Clements and Sarama, 2007, 2008). Operations: Addition and Subtraction The 2- and 3-year-old children can solve change plus/change minus situations and put together/take apart situations with small numbers (to- tals â¤ 5) if the situation is presented with objects or if they are helped to use objects to model these situations (Clements and Sarama, 2007; Fuson, 1988). Children can have experience in learning how to do such adding and subtracting from family members, in child care centers, and from media such as television and CDs. Children may subitize groups of one and two or count these or somewhat larger numbers. To find the total, they may count or put together the subitized quantity into a pattern that is also just seen and not really counted (e.g., two and two make four). Step 2 (Age 4 or Prekindergarten) At this step, children learn to use conceptual subitizing and cardinal counting to solve situation, word, and oral number word problems with totals â¤ 8 and begin to count and to match to find out which set has more or less (see Box 5-10). Cardinal counters at this age level can extend their understanding of relations and of all of the addition/subtraction situations and generalize them to a wider range of settings because their real-world knowledge is more extensive than it was at the previous level. Children can now also count out a specified number of objects, so they can carry out the count all and take away solution methods (Level 1 in Box 5-11) for numbers in their counting accuracy range. They also begin to use counting and matching as well as the earlier perceptual strategies to find which of two sets is more and begin to learn the meaning of the word less. Relations Children at this level continue to use the perceptual strategies they used earlier (general perceptual, length, density) but they can also begin to use matching and counting to find which is less and which is more (see research summarized in Clements and Sarama, 2007, 2008; Fuson, 1988, 1992a, 1992b; Sophian, 1988). However, they can also be easily misled by perceptual cues. For example, the classic tasks used by Piaget (1941/1965) involved two rows of objects in which the objects in one row were moved apart so one row was longer (or occasionally, moved together so one row was shorter). Many children ages 4 and 5 would say that the longer row has

156 MATHEMATICS LEARNING IN EARLY CHILDHOOD more. These children focused either on length or on density, but they could not notice and coordinate both. However, when asked to count in such situations, many 4-year-olds can count both rows accurately, remember both count words, and change them to cardinal numbers and find the order relation on the cardinal numbers (Fuson, 1988). Thus, many 4-year-olds need encouragement to count in more than/less than/equal to situations, especially when the perceptual information is misleading. To use matching successfully to find more than/less than, children may need to learn how to match by drawing lines visually to connect pairs or draw such matching lines if the compared sets are drawn on paper. Then they need to know that the number with any extra objects is more than the other set. It is also helpful to match using actual objects. To use counting successfully, children need to be able to count both sets accurately and remember the first count result while counting the sec- ond set. Here is another example of the need for fluency in counting (see Box 5-1). Without such fluency, some children forget their first count result by the time they have counted the second set. They need more counting practice in such situations. Children also need to know order relations on cardinal numbers. They need to learn the general pattern that most children do derive from the order of the counting words: The number that tells more is farther along (said later) in the number word list than the smaller number (e.g., Fuson, Richards, and Briars, 1982). Activities in which children make sets for both numbers, match them in rows and count them, and discuss the results can help them establish this general pattern. There was an early period in which the counting and matching research had not been done and many researchers and educators suggested that teachers had to wait until children conserved number (said that rows in the classic Piagetian task were equal even in the face of misleading perceptual transformations) to do any real number activities, such as adding and sub- tracting. However, newer research shows that there is a crucial stage for 4- and 5-year-olds in which using counting and matching are important to learn and can lead to correct relational judgments (see the research sum- marized in Clements and Sarama, 2007, 2008; Fuson, 1992a, 1992b). It is true that children typically do not understand that the rows are equal out of a logical necessity until age 6 or 7 (sometimes not until age 8). These older children (ages 6-7) judge the rows to be equal based on mental transforma- tions that they apply to the situation. They do not see the need to count or match after one row is made shorter or longer by moving objects in it together or apart to see that they are equal. They are certain that simply moving the objects in the set does not change the numerosity. This is what Piaget meant by conservation of number. But children can work effectively with situations involving more and less long before they demonstrate this meaning of conservation of number. For progress in relations, it is important that children hear, and try

PATHS FOR NUMBER, RELATIONS, AND OPERATIONS 157 to use, the less common comparative terms such as less, shorter, smaller instead of only hearing or using more, taller, bigger. Initially some children think that less means more because almost all of their experience has been focused on selecting the set with more (e.g., Fuson, Carroll, and Landis, 1996). So children need to hear many examples of fewer and less, although it is not vital that they differentiate these from each other because that is difficult (fewer is used with things you can count, less is used with measured quantities and with numbers). Teachers can also use the comparative terms (for example, bigger and smaller rather than just big and small) so that children gain experience with them, although all children may not become fluent in their use at this level. Operations Problems expressed in words (word problems) can now be solved, although many children may need to act out some word problems in order to understand the meanings of the situation or of some of the words (see research summarized in Fuson 1992a, 1992b). Through such experiences relating actions and words, children gradually extend their vocabulary of words that mean to addâin all, put together, altogether, totalâand of words that mean to subtractâare left, take away, eat, break. Discussing and sharing solutions to word problems and acting out addition/Âsubtraction situations can provide extended experiences for language learning. Children can begin posing such word problems as well as solving them, although many will need help with asking the questions, the most difficult aspect of posing word problems. As with all language learning, it is very important for children to talk and to use the language themselves, so having them retell a word problem in their own words is a powerful general teaching strategy to extend their knowledge and give them practice speaking in English. Drawing the solution actions using circles or other simple shapes in- stead of pictures of real objects can be helpful. The two addends can be separated just by space or encircled separately or separated by a vertical line segment. Some children can also begin to make mathematical drawings to show their solutions. Teacher and child drawings leave a visual record of the full solution that facilitates childrenâs reflecting on the solution, as well as discussing and explaining it. For children, making math drawings is also a creative activity in which they are somehow showing in space actions that occur over time. Children do this in various interesting ways that can lead to productive discussions. Children also become able to use their fingers to add or to subtract using the direct modeling solution methods counting all or taking away (see Box 5-11, Level 1). When counting all, they will count out and raise fingers for the first addend, then for the second addend, and then count all

158 MATHEMATICS LEARNING IN EARLY CHILDHOOD of the raised fingers. (See Box 5-4 for a discussion of different conventions for counting on fingers.) Some children learn at home or in a care center to put the addends on separate hands, while others continue on to the next fingers for the second addend. The former method makes it easier to see the addends, and the lat- ter method makes it easier to see the total. Both methods can be modeled by the teacher. As children become more and more familiar with which group of fingers makes 4 or 5 or 7 fingers, they may not even have to count out the total because they can feel or see the total fingers. Similarly, children using the method of putting fingers on separate hands eventually can just raise the fingers for the addends without counting out the fingers. But they do need initially to count the total. Children who put addends on separate hands may have difficulty with problems with addends over 5 (e.g., 6 + 3) because one cannot put both such numbers on a separate hand. They can, however, continue raising fingers from 6 fingers. Because these problems involve adding 1 or 2, such continuations of 1 and 2 are relatively easy. By now children who have had experience with adding and subtracting situations when they were younger can generalize to solve decontextualized problems that are posed numerically, as in Two and two make how many? (Clements and Sarama, 2007; Fuson, 1988). For some small numbers, chil- dren may have solved such a problem so many times that they know the answer as a verbal statement: Two and two make four. If such knowledge is fluent, children may be able to use it to solve a more complex unknown addend problem. For example, Two and how many make four? Two. For larger numbers, children will need to use objects or fingers to carry out a counting all or taking away solution procedure (Box 5-11) (see research summarized in Fuson 1992a, 1992b). Children will learn new composed/decomposed numerical triads as they have such experiences. The doubles that involve the same addends (2 is 1 and 1, 4 is 2 and 2, 6 is 3 and 3, 8 is 4 and 4) are particularly easy for children to learn because the perceptual and verbal task is simplified by have the same addends (e.g., see research summarized in Fuson, 1992a, 1992b). The visual 5-groups (e.g., 8 is made from 5 and 3) discussed for the number core are also use- ful. Research about powerful patterns for conceptual subitizing for very small numbers would be helpful, including the extent to which flexibility is important beyond a single powerful visual core that will work for all numbers. The put together/take apart situations, and especially the take apart situation, can be used to provide varied numerical experiences with given numbers that help children see all of the addends (partners) hiding inside a given number. For example, children can take apart five to see that it can be made from a three and two and also from four and one. Later on these decomposed/composed triads can be symbolized by equations, such as 5 =

PATHS FOR NUMBER, RELATIONS, AND OPERATIONS 159 3 + 2 and 5 = 4 + 1, giving children experiences with the meaning of the = symbol as is the same number as and with algebraic equations with one number on the left. Initially children shift from seeing the total and then seeing the partners (addends), but with experience and fluency, they can simultaneously see the addend within the total. This is called embedded numbers: The two addends are embedded within the total. Such embedded numbers, along with the number word sequence skill of starting counting at any number, allow children to move to the second level of addition/sub- traction solution procedures, counting on. Initially composed/decomposed number triads and even embedded number triads are constructed with small numbers using conceptual subitizing, but eventually counting is used with larger numbers to construct larger triads. Many children from low-income backgrounds cannot initially solve such oral numerical problems, even with very small numbers (see Chapter 4). They need opportunities to learn and practice the Level 1 solution methods with objects and with fingers and experience composing/Âdecomposing num- bers to be able to see the addends (partners) hiding inside the small numbers 3, 4, 5. Such alternating focusing on the total and then on the partners (addends) will enable them to answer such oral numerical problems and also begin the learning path toward embedded numbers that is vital for the Level 2 addition/subtraction solution methods. Step 3 (Kindergarten) At this step, children extend cardinal counting and use math draw- ings as well as objects to solve situation, word, oral number word, written numeral, and which-is-more/less problems with totals â¤ 10 (see Box 5-10). Written work, including worksheets, is appropriate in kindergarten if it follows up on activities with objects or presents supportive visualizations. Children at these ages need practice that builds fluency after related expe- riences with objects to build mathematical understanding, and they need experience relating symbols for quantities to actual or drawn quantities. Kindergarten children can extend their addition and subtraction prob- lem solving to all problems with totals â¤ 10. Close to half of these prob- lems have one addend of six or more. For these problems, knowing the 5-patterns using fingers for 6 through 10 can be helpful (5 + 1 = 6, 5 + 2 = 7, etc., to 5 + 5 = 10). All children can begin to make math drawings themselves, even for these larger numbers. This allows them to reflect on and discuss their solution methods. Math drawings involving circles or other simple shapes also enable more advanced children to explore prob- lems with totals greater than ten. It is difficult to solve such problems with fingers until one advances to the general counting on solution methods (see Box 5-11, Level 2), which typically does not occur until Grade 1. Children

160 MATHEMATICS LEARNING IN EARLY CHILDHOOD can discuss general patterns they see in addition and subtraction, such as +1 is just the next counting number or â1 is the number just before. Children can discuss adding and subtracting 0 and the pattern it gives: adding or subtracting 0 does not change the original number, so the result (the answer) is the same as the original number. Many children can now informally use the commutative property (A + B = B + A) especially when one number is small (e.g., Baroody and Gannon, 1985; Carpenter et al., 1993; DeCorte and Verschaffel, 1985; for a review of the literature, see Baroody, Wilkins, and Tiilikainen, 2003). Experience with put together ad- dition situations in which the addends do not have different roles provides better support for learning the commutative property than does experience with the change situation (see research described in Clements and Sarama, 2007, 2008; Fuson, 1992a, 1992b) because these addends have such dif- ferent roles in the action. To the child, it actually feels different to have 1 and then get 8 more than to have 8 and get 1 more. It feels better to gain 8 instead of gaining 1, even though you end up with the same amount. In contrast, the numerical work on put together/take apart partners facilitates understanding that the order in which one adds does not matter. Looking at composed/decomposed triads with the same addends also enables children to see and understand commutativity in these examples (for example, see that 9 = 1 + 8 and 9 = 8 + 1 and that the addends are just switched in order but still total the same). All of the work on the relations/operation core in kindergarten serves a double purpose. It helps children solve larger problems and become more fluent in their Level 1 direct modeling solution methods. It also helps them reach fluency with the number word list in addition and subtraction situa- tions, so that the number word list can become a representational tool for use in the counting on solution methods. Different children learn and remember some sums and differences at each level, and it is very useful to know these for small numbers, for ex- ample for totals â¤ 8. But the more important step at the kindergarten level is that children are learning general numerical solution methods that they can extend to larger numbers. Simultaneously they are becoming fluent with these processes and with the number word list, so that they can advance to the Level 2 counting on methods that are needed to solve single-digit sums and differences with totals over ten. Children later in the year can begin to practice the number word list prerequisite for counting on by starting to count at a given number instead of always at one. Kindergarten children are also working on all of the prerequisites for the Level 3 derived fact methods, such as make-a-ten (see Box 5-11). One prerequisite, seeing the tens in teen numbers, was discussed in the number core. The other two prerequisites involve knowing partners of numbers

PATHS FOR NUMBER, RELATIONS, AND OPERATIONS 161 (decomposed/composed numerical triads) to permit flexible breaking apart and combining of numbers to turn them into teen addition or subtraction problems. For example, all of the following addition problemsâ9 + 2, 9 + 3, 9 + 4, . . . , 9 + 9ârequire the same first step: 9 needs 1 more to make ten, so separate the second number into 1 + ?. This triad then becomes 9 + 1 + ? = 10 + ?, which is an easier problem to solve if you know the tens in teen numbers. However, each problem requires a different second step: decomposing the second number to identify the rest of the second addend that will be added to ten (prerequisite 3 for derived facts methods in Box 5-11). For example, 9 + 4 = 9 + 1 + 3 = 10 + 3 = 13, but 9 + 6 = 9 + 1 + 5 = 10 + 5 = 15. So kindergarten children need experiences with finding and learning the partners of various numbers under 10. Childrenâs counting and matching knowledge is now sufficient to ex- tend to relations on sets up through 10 and to more abstract ways of presenting such relational situations as two rows of drawings that can be matched by drawing lines connecting them. As discussed above for Step 2, children will be more accurate when these objects are already matched instead of being visually misleading (for example, the longer row has less). They therefore can start with the simpler nonmisleading situations and extend to the visually misleading situations when they have mastered such matched situations. Again, differentiating length and number meanings of more will be helpful (which looks like more and which really is more). Children who have not had sufficient experiences matching objects at Step 2 will need such experiences to support the more advanced activities in which matching is done by drawing lines. Working with the terms more and less can also be an opportunity to discuss and emphasize that length units used in measuring a length must touch each other and cover the whole length from beginning to end to get an accurate length measurement. But things children are counting can be spread apart or moved around and they will still have the same number of things. Comparing objects spaced evenly in two rows can also be related to picture graphs, which record numbers of different kinds of data as a row of the same pictures (see the Chapter 2 discussion in the Mathemati- cal Connections section). Activities in which children compare two rows of drawings by counting or matching them can be considered as using picture graphs if each drawing in one row is the same. What is important about such activities is that children talk about them using comparison language (There are more suns than clouds or There are fewer clouds than suns) and describe how they found their answer. Children at this level can also prepare for the comparison problems at Grade 1 by beginning to equalize two related sets. For example, for a row of 5 above a row of 7, they can be asked to add more to the row of 5 to

162 MATHEMATICS LEARNING IN EARLY CHILDHOOD make it equal to the row of 7 and write their addition 5 + 2. This 2 is the difference between 5 and 7, it is the amount extra 7 has, so such exercises help children begin to see this third quantity in the comparison situation. Writing Equations There is not sufficient evidence to indicate the best time for teachers to start writing addition and subtraction problems in equations or for students to do so. The equation form can be confusing to some students even in Grade 1, and students may confuse the symbols + = and â. This confusion and limited meanings for the = sign often continue for many years and are of concern for the later learning of algebra. Because the fundamental aspect of an equation is that the sides are equal to each other, it is important for children to learn to conceptually chunk each side. Thus, some children may need extensive experience just with expressions, such as 3 + 2 or 7 â 5, before these are used in equations. These forms might be introduced before the full equation is introduced, perhaps even with 4-year-olds. It may also help for the teacher to circle or underline these expressions to indicate that this group of symbols is a chunk that represents a single number. Future research directed at such issues of when and how to write such pre-equation forms would be helpful. The other issue with equations is the form of the equation to write. As mentioned earlier, it is important for later algebraic understanding of acceptable forms of equations for children to see equations with only one number on the left, such as 6 = 4 + 2 to show that 6 breaks apart to make 4 and 2. This equation form can be written for take apart situations in which the total is being separated into two parts, for example, Grammy has 6 flowers. She put four flowers in one vase and two flowers in the other vase. Children can show this situation with objects or fingers (Count out 6 objects and then separate them into 4 and 2) or make a math drawing of it while the teacher records the situation in an equation. This form can also be used in practice activities with objects in which children find all of the partners (addends) of a given number. For example, children can make 5 using two different colors of objects, and each color can show the partners. The teacher can record all of the partners that children find: 5 = 1 + 4, 5 = 2 + 3, 5 = 3 + 2, 5 = 4 + 1. This can be in a situation (Letâs find all of the ways that Grammy can put her 5 flowers in her 2 vases) or just an activity with numbers (Letâs find all of the partners of 5). Change plus and change minus situations can be recorded by equations with only one number on the right because that is the action in these situa- tions (see Box 2-4), for example, 3 + 1 = 4 or 5 â 2 = 3. In these equations the = sign is really more like an arrow, meaning gives or results in. As dis-

PATHS FOR NUMBER, RELATIONS, AND OPERATIONS 163 cussed, this is often the only meaning of = that students in the United States know, and this interferes with their use of algebra. So it is really important that they also see and use forms like 5 = 3 + 2 to show the numbers hiding inside a number, the partners (addends) that make that number. Step 4 (Grade 1) At this step, children build on their earlier number and relations/Â operation knowledge and skills to advance to Level 2 counting on solu- tion methods. They also come to understand that addition is related to subtraction and can think of subtraction as finding an unknown addend (see Box 5-10). Grade 1 addition and subtraction is the culmination of all of the num- ber core and relations/operation core experiences and expertise that have been building since birth, for those who have been given sufficient opportu- nities to build such competence. Foundational and achievable relations and operations content for Grade 1 children is summarized in Box 5-9. For all of the earlier experiences to come together into the Level 2 counting on solution methods, some children may still need some targeted practice in beginning counting at any number instead of always starting at one (one of the prerequisites for counting on). It is also helpful to begin counting on in some kind of structured visual setting, so that children can conceptualize the relationships between the counting and cardinal meanings of number words. Counting on is not a rote method. It requires a shift in word meaning for the first addend from its cardinal meaning of the number in that first addend to a counting meaning, as children count on from that first addend to the total. Children then must shift from that last counted word to its car- dinal meaning of how many objects there are in total. For example, seeing circles for both addends in a row with the problem printed above enables children to count both addends and then count all to find the total (their usual Level 1 direct modeling solution method). But after several times of counting all, they can be asked what number they say when they count the last circle in the group of 6 and whether they need to count all of the objects or could they just start at 6. Going back and forth between this counting on and the usual counting all enables children to see that counting on is just an abbreviation of counting all, in which the initial counts are omitted (e.g., Fuson, 1982; Fuson and Secada, 1986; Secada, Fuson, and Hall, 1983). 6 + 3 is a cardinal number. Six o o o o o oâ o o o 1 2 3 4 5 6â 7 8 9 here is a count number when counting all. Six

164 MATHEMATICS LEARNING IN EARLY CHILDHOOD siiiixxxâ 7 8 9 count on like this, a child must shift from the To 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 addendâof 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 3 circles and another 3 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

PATHS FOR NUMBER, RELATIONS, AND OPERATIONS 165 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 â 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 five 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 9 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 C Â lements 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.

166 MATHEMATICS LEARNING IN EARLY CHILDHOOD Issues in Learning Relations and Operations The Extensive 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 âmemorizing the factsâ or ârecalling the facts,â 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.

PATHS FOR NUMBER, RELATIONS, AND OPERATIONS 167 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 unitsâthey 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

168 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 modelsâeach square is an object that can be countedâso these are appropriate for children from age 2 through Grade 1. Some research sum- marized in Chapter 3 did focus on childrenâs and adultâs 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 childrenâs 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 Childrenâs 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 childrenâs 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-

PATHS FOR NUMBER, RELATIONS, AND OPERATIONS 169 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 knowledgeâfor 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). Childrenâs difficulties in subtraction: Some causes and questions. Journal for Research in Mathematics Education, 15(3), 203-213. Baroody, A.J. (1987). Childrenâs Mathematical Thinking: A Developmental Framework for Pre- school, Primary, and Special Education Teachers. New York: Teachers College Press. Baroody, A.J. (1992). The development of preschoolersâ 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 Childrenâs Mathematical Power: An Inves- tigative Approach to K-8 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 n Â umerical-writing difficulty. Teaching Exceptional Children, 25(3), 14-16. Baroody, A.J., Wilkins, J.L.M., and Tiilikainen, S.H. (2003). The development of childrenâs understanding of additive commutativity: From protoquantitative concept to general con- cept? In A.J. Baroody and A. Dowker (Eds.), The Development of Arithmetic Concepts and Skills: Constructing Adaptive Expertise (pp. 127-160). Mahwah, NJ: Erlbaum.

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