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257 FOSTERING THE DEVELOPMENT OF WHOLE-NUMBER SENSE 6 Fostering the Development of Whole-Number Sense: Teaching Mathematics in the Primary Grades Sharon Griffin After 15 years of inquiry into children’s understanding and learning of whole numbers, I can sum up what I have learned very simply. To teach math, you need to know three things. You need to know where you are now (in terms of the knowledge children in your classroom have available to build upon). You need to know where you want to go (in terms of the knowledge you want all children in your classroom to acquire during the school year). Finally, you need to know what is the best way to get there (in terms of the learning opportunities you will provide to enable all children in your class to achieve your stated objectives). Although this sounds simple, each of these points is just the tip of a large iceberg. Each raises a question (e.g., Where are we now?) that I have come to believe is crucial for the design of effective mathematics instruction. Each also points to a body of knowledge (the iceberg) to which teachers must have access in order to answer that question. In this chapter, I explore each of these icebergs in turn in the context of helping children in the primary grades learn more about whole numbers. Readers will recognize that the three things I believe teachers need to know to teach mathematics effectively are similar in many respects to the knowledge teachers need to implement the three How People Learn prin- ciples (see Chapter 1) in their classrooms. This overlap should not be sur- prising. Because teaching and learning are two sides of the same coin and
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258 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM because effective teaching is defined primarily in terms of the learning it supports, we cannot talk about one without talking about the other. Thus when I address each of the three questions raised above, I will at the same time offer preschool and elementary mathematics teachers a set of resources they can use to implement the three principles of How People Learn in their classrooms and, in so doing, create classrooms that are student-centered, knowledge-centered, community-centered, and assessment-centered. Addressing the three principles of How People Learn while exploring each question occurs quite naturally because the bodies of knowledge that underlie effective mathematics teaching provide a rich set of resources that teachers can use to implement these principles in their classrooms. Thus, when I explore question 1 (Where are we now?) and describe the number knowledge children typically have available to build upon at several specific age levels, I provide a tool (the Number Knowledge test) and a set of ex- amples of age-level thinking that teachers can use to enact Principle 1— eliciting, building upon, and connecting student knowledge—in their class- rooms. When I explore question 2 (Where do I want to go?) and describe the knowledge networks that appear to be central to children’s mathematics learning and achievement and the ways these networks are built in the normal course of development, I provide a framework that teachers can use to enact Principle 2—building learning paths and networks of knowledge— in their classrooms. Finally, when I explore question 3 (What is the best way to get there?) and describe elements of a mathematics program that has been effective in helping children acquire whole-number sense, I provide a set of learning tools, design principles, and examples of classroom practice that teachers can use to enact Principle 3—building resourceful, self-regulating mathematical thinkers and problem solvers—in their classrooms. Because the questions I have raised are interrelated, as are the principles themselves, teaching practices that may be effective in answering each question and in promoting each principle are not limited to specific sections of this chapter, but are noted throughout. I have chosen to highlight the questions themselves in my introduction to this chapter because it was this set of questions that motivated my inquiry into children’s knowledge and learning in the first place. By asking this set of questions every time I sat down to design a math lesson for young chil- dren, I was able to push my thinking further and, over time, construct better answers and better lessons. If each math teacher asks this set of questions on a regular basis, each will be able to construct his or her own set of answers for the questions, enrich our knowledge base, and improve math- ematics teaching and learning for at least one group of children. By doing so, each teacher will also embody the essence of what it means to be a resourceful, self-regulating mathematics teacher. The questions themselves are thus more important than the answers. But the reverse is also true:
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259 FOSTERING THE DEVELOPMENT OF WHOLE-NUMBER SENSE although good questions can generate good answers, rich answers can also generate new and better questions. I now turn to the answers I have found useful in my own work with young children. By addressing question 2 (Where do I want to go?) first, I hope to give readers a sense of the general direction in which we are head- ing before I turn to question 1 (Where are we now?) and provide a detailed description of the knowledge children generally have available to build upon at each age level between 4 and 8. While individual children differ a great deal in the rate at which they acquire number knowledge, teachers are charged with teaching a class of students grouped by age. It is therefore helpful in planning instruction to focus on the knowledge typical among children of a particular age, with the understanding that there will be consid- erable variation. In a subsequent section, I use what we have learned about children’s typical age-level understandings to return to the issue of the knowl- edge to be taught and to provide a more specific answer for question 2. DECIDING WHAT KNOWLEDGE TO TEACH All teachers are faced with a dizzying array of mathematics concepts and skills they are expected to teach to groups of students who come to their classrooms with differing levels of preparedness for learning. This is true even at the preschool level. For each grade level, the knowledge to be taught is prescribed in several documents—the national standards of the National Council of Teachers of Mathematics (NCTM), state and district frame- works, curriculum guides—that are not always or even often consistent. Deciding what knowledge to teach to a class as a whole or to any individual child in the class is no easy matter. Many primary school teachers resolve this dilemma by selecting number sense as the one set of understandings they want all students in their class- rooms to acquire. This makes sense in many respects. In the NCTM stan- dards, number sense is the major learning objective in the standard (num- bers and operations) to which primary school teachers are expected to devote the greatest amount of attention. Teachers also recognize that children’s ability to handle problems in other areas (e.g., algebra, geometry, measure- ment, and statistics) and to master the objectives listed for these standards is highly dependent on number sense. Moreover, number sense is given a privileged position on the report cards used in many schools, and teachers are regularly required to evaluate the extent to which their students “demon- strate number sense.” In one major respect, however, the choice of number sense as an instructional objective is problematic. Although most teachers and lay people alike can easily recognize number sense when they see it, defining what it is and how it can be taught is much more difficult.
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260 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM Consider the responses two kindergarten children provide when asked the following question from the Number Knowledge test (described in full later in this chapter): “If you had four chocolates and someone gave you three more, how many would you have altogether?” Alex responds by scrunching up his brow momentarily and saying, “seven.” When asked how he figured it out, he says, “Well, ‘four’ and ‘four’ is ‘eight’ [displaying four fingers on one hand and four on the other hand to demonstrate]. But we only need three more [taking away one finger from one hand to demonstrate]. So I went—‘seven,’ ‘eight.’ Seven is one less than eight. So the answer is seven.” Sean responds by putting up four fingers on one hand and saying (under his breath), “Four. Then three more—‘five, six, seven.’” In a normal tone of voice, Sean says “seven.” When asked how he figured it out, Sean is able to articulate his strategy, saying, “I started at four and counted—‘five, six, seven’” (tapping the table three times as he counts up, to indicate the quantity added to the initial set). It will be obvious to all kindergarten teachers that the responses of both children provide evidence of good number sense. The knowledge that lies behind that sense may be much less apparent, however. What knowledge do these children have that enables them to come up with the answer in the first place and to demonstrate number sense in the process? Scholars have studied children’s mathematical thinking and problem solving, tracing the typical progression of understanding or developmental pathway for acquir- ing number knowledge.1 This research suggests that the following under- standings lie at the heart of the number sense that 5-year-olds such as Alex and Sean are able to demonstrate on this problem: (1) they know the count- ing sequence from “one” to “ten” and the position of each number word in the sequence (e.g., that “five” comes after “four” and “seven” comes before “eight”); (2) they know that “four” refers to a set of a particular size (e.g., it has one fewer than a set of five and one more than a set of 3), and thus there is no need to count up from “one” to get a sense of the size of this set; (3) they know that the word “more” in the problem means that the set of four chocolates will be increased by the precise amount (three chocolates) given in the problem; (4) they know that each counting number up in the count- ing sequence corresponds precisely to an increase of one unit in the size of a set; and (5) it therefore makes sense to count on from “four” and to say the next three numbers up in the sequence to figure out the answer (or, in Alex’s case, to retrieve the sum of four plus four from memory, arrive at “eight,” and move one number back in the sequence). This complex knowl-
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261 FOSTERING THE DEVELOPMENT OF WHOLE-NUMBER SENSE edge network—called a central conceptual structure for whole number—is described in greater detail in a subsequent section. The knowledge that Alex and Sean demonstrate is not limited to the understandings enumerated above. It includes computational fluency (e.g., ease and proficiency in counting) and awareness of the language of quantity (e.g., that “altogether” indicates the joining of two sets), which were ac- quired earlier and provided a base on which the children’s current knowl- edge was constructed. Sean and Alex also demonstrate impressive metacognitive skills (e.g., an ability to reflect on their own reasoning and to communicate it clearly in words) that not only provide evidence of number sense, but also contributed to its development. Finally, children who demonstrate this set of competencies also show an ability to answer questions about the joining of two sets when the con- texts vary considerably, as in the following problems: “If you take four steps and then you take three more, how far have you gone?” and “If you wait four hours and then you wait three more, how long have you waited?” In both of these problems, the quantities are represented in very different ways (as steps along a path, as positions on a dial), and the language used to describe the sum (“How far?” “How long?”) differs from that used to describe the sum of two groups of objects (“How many?”). The ability to apply num- ber knowledge in a flexible fashion is another hallmark of number sense. Each of the components of number sense mentioned thus far is de- scribed in greater detail in a subsequent section of this chapter. For now it is sufficient to point out that the network of knowledge the components repre- sent—the central conceptual structure for whole number—has been found to be central to children’s mathematics learning and achievement in at least two ways. First, as mentioned above, it enables children to make sense of a broad range of quantitative problems in a variety of contexts (see Box 6-1 for a discussion of research that supports this claim). Second, it provides the base—the building block—on which children’s learning of more complex number concepts, such as those involving double-digit numbers, is built (see Box 6-2 for research support for this claim). Consequently, this network of knowledge is an important set of understandings that should be taught. In choosing number sense as a major learning goal, teachers demonstrate an intuitive understanding of the essential role of this knowledge network and the importance of teaching a core set of ideas that lie at the heart of learning and competency in the discipline (learning principle 2). Having a more explicit understanding of the factual, procedural, and conceptual under- standings that are implicated and intertwined in this network will help teachers realize this goal for more children in their classrooms. Once children have consolidated the set of understandings just described for the oral counting sequence from “one” to “ten,” they are ready to make sense of written numbers (i.e., numerals). Now, when they are exposed to
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262 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM The Central Conceptual Structure Hypothesis: BOX 6-1 Support for the First Claim A central conceptual structure is a powerful organizing knowledge net- work that is extremely broad in its range of application and that plays a central role in enabling individuals to master the problems that the domain presents. The word “central” implies (1) that the structure is vital to suc- cessful performance on a range of tasks, ones that often transcend indi- vidual disciplinary boundaries; and (2) that future learning in these tasks is dependent on the structure, which often forms the initial core around which all subsequent learning is organized. To test the first of these claims, Griffin and Case selected two groups of kindergarten children who were at an age when children typically have acquired the central conceptual structure for whole number, but had not yet done so.2 All the children were attending schools in low-income, in- ner-city communities. In the first part of the kindergarten year, all the chil- dren were given a battery of developmental tests to assess their central conceptual understanding of whole number (Number Knowledge test) and their ability to solve problems in a range of other areas that incorporate number knowledge, including scientific reasoning (Balance Beam test), social reasoning (Birthday Party task), moral reasoning (Distributive Jus- tice task), time telling (Time test), and money knowledge (Money test). On this test administration, no child in either group passed the Number Knowledge test, and fewer than 20 percent of the children passed any of the remaining tests. One group of children (the treatment group) was exposed to a math- ematics program called Number Worlds that had been specifically designed to teach the central conceptual structure for whole number. The second group of children (a matched control group) received a variety of other forms of mathematics instruction for the same time period (about 10 weeks). The performance of these two groups on the second administra- the symbols that correspond to each number name and given opportunities to connect name to symbol, they will bring all the knowledge of what that name means with them, and it will accrue to the symbol. They will thus be able to read and write number symbols with meaning. To build a learning path that matches children’s observed progression of understanding, this would be a reasonable next step for teachers to take. Finally, with experi- ence in using this knowledge network, children eventually become capable
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263 FOSTERING THE DEVELOPMENT OF WHOLE-NUMBER SENSE tion of the same tests at the end of the kindergarten year is presented in the following table. The treatment group—those exposed to the Number Worlds curriculum—improved substantially in all test areas, far surpass- ing the performance of the control group. Because no child in the treat- ment group had received any training in any of the areas tested in this battery besides number knowledge, the strong post-training performance of the treatment group on these tasks can be attributed to the construc- tion of the central conceptual structure for whole number, as demonstrated in the children’s (post-training) performance on the Number Knowledge test. Other factors that might have accounted for these findings, such as more individual attention and/or instructional time given to the treatment group, were carefully controlled in this study. Percentages of Children Passing the Second Administration of the Number Knowledge Test and Five Numerical Transfer Tests ________________________________________________________________________ Control Group Treatment Group Testa (N = 24) (N = 23) _________________________________________________________________________ Number Knowledge (5/6) 25 87 Balance Beam (2/2) 42 96 Birthday Party (2/2) 42 96 Distributive Justice (2/2) 37 87 Time Telling (4/5) 21 83 Money Knowledge (4/6) 17 43 aNumber of items out of total used as the criterion for passing the test are given in parentheses. of applying their central conceptual understandings to two distinct quantita- tive variables (e.g., tens and ones, hours and minutes, dollars and cents) and of handling two quantitative variables in a coordinated fashion. This ability permits them to solve problems involving double-digit numbers and place value, for example, and introducing these concepts at this point in time (sometime around grade 2) would be a reasonable next step for teachers to
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264 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM The Central Conceptual Structure Hypothesis: Support for BOX 6-2 the Second Claim To test the second centrality claim—that future learning is dependent on the acqui- sition of the central conceptual structure for whole number—Griffin and Case con- ducted a follow-up study using the same sample of children as that in Box 6-1.3 Children in both the treatment and control groups had graduated to a variety of first-grade classrooms in a number of different schools. Those who had remained in the general geographic area were located 1 year later and given a range of as- sessments to obtain measures of their mathematics learning and achievement in grade 1. Their teachers, who were blind to the children’s status in the study, were also asked to rate each child in their classroom on a number of variables. The results, displayed in the following table, present an interesting portrait of the importance of the central conceptual structure (assessed by performance at the 6-year-old level of the Number Knowledge test) for children’s learning and achievement in grade 1. Recall that 87 percent of the treatment group had passed this level of the number knowledge test at the end of kindergarten compared with 25 percent of the control group. As the table indicates, most of the children in the control group (83 percent) had acquired this knowledge by the end of grade 1, but it appears to have been too late to enable many of them to master the grade 1 arithmetic tasks that require conceptual understanding (e.g., the Oral Arithmetic test; the Word Problems; test and teacher ratings of number sense, number mean- ings, and number use). On all of these measures, children who had acquired the central conceptual structure before the start of the school year did significantly better. On the more traditional measures of mathematics achievement (e.g., the Written Arithmetic test and teacher ratings of addition and subtraction) that rely more on procedural knowledge than conceptual understanding, the performance of children in the control group was stronger. It was still inferior, however, in abso- lute terms to the performance of children in the treatment group. Possibly the most interesting finding of all is the difference between the two groups on tests that tap knowledge not typically taught until grade 2 (e.g., the 8- year-old level of the Number Knowledge test and the 8-year-old level of the Word Problems test). On both of these tests, a number of children in the treatment group demonstrated that they had built upon their central conceptual structure for whole number during their first-grade experience and were beginning to construct the more elaborate understandings required to mentally solve double-digit arithmetic problems. Few children in the control group demonstrated this level of learning.
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265 FOSTERING THE DEVELOPMENT OF WHOLE-NUMBER SENSE Percentages of Children Passing the Number Knowledge Test and Measures of Arithmetic Learning and Achievement at the End of Grade 1 Control Treatment Group Group Significance of differencea Test (N = 12) (N= 11) Number Knowledge Test 6-year-old level 83 100 ns a 8-year-old level 0 18 a Oral Arithmetic Test 33 82 Written Arithmetic Test 75 91 ns Word Problems Test a 6-year-old level 54 96 a 8-year-old level 13 46 Teacher Rating a Number sense 24 100 a Number meaning 42 88 a Number use 42 88 Addition 66 100 ns Subtraction 66 100 ns a ns= not significant; = significant at the .01 level or better.
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266 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM take in building learning paths that are finely attuned to children’s observed development of number knowledge. In this brief example, several developmental principles that should be considered in building learning paths and networks of knowledge (learning principle 2) for the domain of whole numbers have come to light. They can be summarized as follows: • Build upon children’s current knowledge. This developmental prin- ciple is so important that it was selected as the basis for one of the three primary learning principles (principle 1) of How People Learn. • Follow the natural developmental progression when selecting new knowledge to be taught. By selecting learning objectives that are a natural next step for children (as documented in cognitive developmental research and described in subsequent sections of this chapter), the teacher will be creating a learning path that is developmentally appropriate for children, one that fits the progression of understanding as identified by researchers. This in turn will make it easier for children to construct the knowledge network that is expected for their age level and, subsequently, to construct the higher-level knowledge networks that are typically built upon this base. • Make sure children consolidate one level of understanding before moving on to the next. For example, give them many opportunities to solve oral problems with real quantities before expecting them to use formal sym- bols. • Give children many opportunities to use number concepts in a broad range of contexts and to learn the language that is used in these contexts to describe quantity. I turn now to question 1 and, in describing the knowledge children typically have available at several successive age levels, paint a portrait of the knowledge construction process uncovered by research—the step-by- step manner in which children construct knowledge of whole numbers between the ages of 4 and 8 and the ways individual children navigate this process as a result of their individual talent and experience. Although this is the subject matter of cognitive developmental psychology, it is highly rel- evant to teachers of young children who want to implement the develop- mental principles just described in their classrooms. Because young chil- dren do not reflect on their own thinking very often or very readily and because they are not skilled in explaining their reasoning, it is difficult for a teacher of young children to obtain a picture of the knowledge and thought processes each child has available to build upon. The results of cognitive developmental research and the tools that researchers use to elicit children’s understandings can thus supplement teachers’ own knowledge and exper- tise in important ways, and help teachers create learner-centered class- rooms that build effectively on students’ current knowledge. Likewise, hav-
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267 FOSTERING THE DEVELOPMENT OF WHOLE-NUMBER SENSE ing a rich picture of the step-by step manner in which children typically construct knowledge of whole numbers can help teachers create knowl- edge-centered classrooms and learning pathways that fit children’s sponta- neous development. BUILDING ON CHILDREN’S CURRENT UNDERSTANDINGS What number knowledge do children have when they start preschool around the age of 4? As every preschool teacher knows, the answer varies widely from one child to the next. Although this variation does not disap- pear as children progress through the primary grades, teachers are still re- sponsible for teaching a whole classroom of children, as well as every child within it, and for setting learning objectives for their grade level. It can be a great help to teachers, therefore, to have some idea of the range of under- standings they can expect for children at their grade level and, equally im- portant, to be aware of the mistakes, misunderstandings, and partial under- standings that are also typical for children at this age level. To obtain a portrait of these age-level understandings, we can consider the knowledge children typically demonstrate at each age level between ages 4 and 8 when asked the series of oral questions provided on the Num- ber Knowledge test (see Box 6-3). The test is included here for discussion purposes, but teachers who wish to use it to determine their student’s cur- rent level of understanding can do so. Before we start, a few features of the Number Knowledge test deserve mention. First, because this instrument has been called a test in the develop- mental research literature, the name has been preserved in this chapter. However, this instrument differs from school tests in many ways. It is admin- istered individually, and the questions are presented orally. Although right and wrong answers are noted, children’s reasoning is equally important, and prompts to elicit this reasoning (e.g., How do you know? How did you figure that out?) are always provided on a subset of items on the test, espe- cially when children’s thinking and/or strategy use is not obvious when they are solving the problems posed. For these reasons, the “test” is better thought of as a tool or as a set of questions teachers can use to elicit children’s conceptions about number and quantity and to gain a better understanding of the strategies children have available to solve number problems. When used at the beginning (and end) of the school year, it provides a good picture of children’s entering (and exit) knowledge. It also provides a model for the ongoing, formative assessments that are conducted throughout the school year in assessment-centered classrooms. Second, as shown in Box 6-3, the test is divided into three levels, with a preliminary (warm-up) question. The numbers associated with each level
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298 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM their heads when spill cards (e.g., – 4) are added to the set of cards in the well pile. When they encounter more-powerful dragons whose fire can be extinguished only with 20 buckets of water, they become capable of per- forming these operations with larger sets of numbers and with higher num- bers. When they are required to submit formal proof to the mayor of the village that they have amassed sufficient pails of water to put out the dragon’s fire before they are allowed to do so, they become capable of writing a series of formal expressions to record the number of pails received and spilled over the course of the game. In such contexts, children have ample opportunity to use the formal symbol system in increasingly efficient ways to make sense of quantitative problems they encounter in the course of their own activity. Design Principle 5: Providing Opportunities for Children to Acquire Computational Fluency As Well As Conceptual Understanding Although opportunities to acquire computational fluency as well as con- ceptual understanding are built into every Number Worlds activity, compu- tational fluency is given special attention in the activities developed for the Warm-Up period of each lesson. In the prekindergarten and kindergarten programs, these activities typically take the form of count-up and count- down games that are played in each land, with a prop appropriate for that land. This makes it possible for children to acquire fluency in counting and, at the same time, to acquire a conceptual understanding of the changes in quantity that are associated with each successive number up (or down) in the counting sequence. This is illustrated in an activity, developed for Sky Land, that is always introduced after children have become reasonably flu- ent in the count-up activity that uses the same prop. Sky Land Blastoff In this activity, children view a large, specially designed thermometer with a moveable red ribbon that is set to 5 (or 10, 15, or 20, depending on children’s competence) (see Figure 6-9). Children pretend to be on a rocket ship and count down while the teacher (or a child volunteer) moves the red ribbon on the thermometer to correspond with each number counted. When the counting reaches “1,” all the children jump up and call “Blastoff!” The sequence of counting is repeated if a counting mistake is made or if anyone jumps up too soon or too late. The rationale that motivated this activity is as follows: “Seeing the level of red liquid in a thermometer drop while count- ing down will give children a good foundation for subtraction by allowing
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299 FOSTERING THE DEVELOPMENT OF WHOLE-NUMBER SENSE FIGURE 6-9 A specially designed thermometer for the Sky Land Blastoff activity—to provide an understanding of the changes in quantity associated with each successive number (up) or down in the counting sequence. them to see that a quantity decreases in scale height with each successive number down in the sequence. This will also lay a foundation for measure- ment” (Sky Land: Activity #2). This activity is repeated frequently over the course of the school year, with the starting point being adjusted over time to accommodate children’s growing ability. Children benefit immensely from opportunities to perform (or lead) the count-down themselves and/or to move the thermometer rib- bon while another child (or the rest of the class) does the counting. When children become reasonably fluent in basic counting and in serial counting (i.e., children take turns saying the next number down), the teacher adds a level of complexity by asking them to predict where the ribbon will be if it is on 12, for example, and they count down (or up) two numbers, or if it is on 12 and the temperature drops (or rises) by 2 degrees. Another form of complexity is added over the course of the school year when children are asked to demonstrate another way (e.g., finger displays, position on a hu- man game mat) to represent the quantity depicted on the thermometer and the way this quantity changes as they count down. By systematically increas- ing the complexity of these activities, teachers expose children to a learning path that is finely attuned to their growing understanding (learning principle 1) and that allows them to gradually construct an important network of conceptual and procedural knowledge (learning principle 2). In the programs for first and second grade, higher-level computation skills (e.g., fluent use of strategies and procedures to solve mental arithmetic
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300 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM problems) are fostered in the Warm-Up activities. In Guess My Number, for example, the teacher or a child picks a number card and, keeping it hidden, generates two clues that the rest of the class can use to guess the number (e.g., it is bigger than 25 and smaller than 29). Guessers are allowed to ask one question, if needed, to refine their prediction (e.g., “Is it an odd num- ber?” “Is it closer to 25 or to 29?”). Generating good clues is, of course, more difficult than solving the prob- lem because doing so requires a refined sense of the neighborhood of num- bers surrounding the target number, as well as their relationship to this number. In spite of the challenges involved, children derive sufficient enjoy- ment from this activity to persevere through the early stages and to acquire a more refined number sense, as well as greater computational fluency, in the process. In one lovely example, a first-grade student provided the fol- lowing clues for the number he had drawn: “It is bigger than 8 and it is 1 more than 90 smaller than 100.” The children in the class were stymied by these clues until the teacher unwittingly exclaimed, “Oh, I see, you’re using the neighborhood number line,” at which point all children followed suit, counted down 9 blocks of houses, and arrived at a correct prediction, “9.” Design Principle 6: Encouraging the Use of Metacognitive Processes (e.g., Problem Solving, Communication, Reasoning) That Will Facilitate Knowledge Construction In addition to opportunities for problem solving, communication, and reasoning that are built into the activities themselves (as illustrated in the examples provided in this chapter), three additional supports for these pro- cesses are included in the Number Worlds program. The first is a set of question cards developed for specific stages of each small-group game. The questions (e.g., “How many buckets of water do you have now?”) were designed to draw children’s attention to the quantity displays they create during game play (e.g., buckets of water collected and spilled) and the changes in quantity they enact (e.g., collecting four more buckets), and to prompt them to think about these quantities and describe them, performing any computations necessary to answer the question. Follow-up questions that are also included (e.g., “How did you figure that out?”) prompt children to reflect on their own reasoning and to put it into words, using the lan- guage of mathematics to do so. Although the question cards are typically used by the teacher (or a teacher’s aide) at first, children can gradually take over this function and, in the process, take greater control over their own learning (learning principle 3). This transition is facilitated by giving one child in the group the official role of Question Poser each time the game is
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301 FOSTERING THE DEVELOPMENT OF WHOLE-NUMBER SENSE played. By giving children important roles in the learning process (e.g., Question-Poser, Facilitator, Discussion Leader, Reporter) and by allowing them to be teachers as well as learners, teachers can create the sort of com- munity-centered classroom that is described in Chapters 1 and 5. The second support is a set of dialogue prompts included in the teacher’s guide, which provides a more general set of questions (e.g., “Who has gone the farthest? How do you know?”) than those provided with the game. Al- though both sets of questions are highly useful in prompting children to use metacognitive processes to make mathematical sense of their own activity, they provide no guidance on how a teacher should respond to the answers children provide. Scaffolding good math talk is still a significant challenge for most primary and elementary teachers. Having a better understanding of the sorts of answers children give at different age levels, as well as increased opportunities to listen to children explain their thinking, can be helpful in building the expertise and experience needed for the exceedingly difficult task of constructing follow-up questions for children’s answers that will push their mathematical thinking to higher levels. The third support for metacognitive processes that is built into the Num- ber Worlds program is a Wrap-Up period that is provided at the end of each lesson. In Wrap-Up, the child who has been assigned the role of Reporter for the small-group problem-solving portion of the lesson (e.g., game play) describes the mathematical activity his or her group did that day and what they learned. The Reporter then takes questions from the rest of the class, and any member of the Reporter’s team can assist in providing answers. It is during this portion of the lesson that the most significant learning occurs because children have an opportunity to reflect on aspects of the number system they may have noticed during game play, explain these concepts to their peers, and acquire a more explicit understanding of the concepts in the process. Over time, Wrap-Up comes to occupy as much time in the math lesson as all the preceding activities (i.e., the Warm-Up activities and small- group problem-solving activities) put together. With practice in using this format, teachers become increasingly skilled at asking good questions to get the conversation going and, immediately thereafter, at taking a back seat in the discussion so that children have ample opportunity to provide the richest answers they are capable of gen- erating at that point in time. (Some wonderful examples of skilled teachers asking good questions in elementary mathematics classrooms are available in the video and CD-ROM products of the Institute for Learning [www.institutefor learning.org].) This takes patience, a willingness to turn control of the discussion over to the children, and faith that they have something important to say. Even at the kindergarten level, children appear to be better equipped to rise to this challenge than many teachers, who, having been taught that they should assume the leadership role in the class,
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302 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM often feel that they should dominate the discussion. Teachers who can rise to this challenge have found that their faith is amply rewarded by the so- phistication of the explanations children provide, even at the kindergarten level; by the opportunities this occasion provides for assessing children’s growth and current understandings; and by the learning and achievement gains children demonstrate on standard measures. WHAT SORTS OF LEARNING DOES THIS APPROACH MAKE POSSIBLE? The Number Worlds program was developed to address three major learning goals: to enable children to acquire (1) conceptual knowledge of number as well as procedural knowledge (e.g., computational fluency); (2) number sense (e.g., an ability to use benchmark values, an ability to solve problems in a range of contexts); and (3) an interest in and positive attitude toward mathematics. Program evaluation for the most part has focused on assessing the extent to which children who have been exposed to the pro- gram have been able to demonstrate gains on any of these fronts. The re- sults of several evaluation studies are summarized below. The Number Worlds program has now been tried in several different communities in Canada and in the United States. For research purposes, the groups of students followed have always been drawn from schools serving low-income, predominantly inner-city communities. This decision was based on the assumption that if the program works for children known to be at risk for school failure, there is a good chance that it will work as well, or even better, for those from more affluent communities. Several different forms of evaluation have been conducted. In the first form of evaluation, children who had participated in the kindergarten level of the program (formerly called Rightstart) were com- pared with matched controls who had taken part in a math readiness pro- gram of a different sort. On tests of mathematical knowledge, on a set of more general developmental measures, and on a set of experimental mea- sures of learning potential, children who had participated in the Number Worlds program consistently outperformed those in the control groups (see Box 6-1 for findings from one of these studies).14 In a second type of evalu- ation, children who had taken part in the kindergarten level of the program (and who had graduated into a variety of more traditional first-grade class- rooms) were followed up 1 year later and evaluated on an assortment of mathematical and scientific tests, using a double-blind procedure. Once again, those who had participated in the Number Worlds program in kindergarten were found to be superior on virtually all measures, including teacher evalu- ations of “general number sense” (see Box 6-2).15
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303 FOSTERING THE DEVELOPMENT OF WHOLE-NUMBER SENSE The expansion of the Number Worlds program to include curricula for first and second grades permitted a third form of evaluation—a longitudinal study in which children were tracked over a 3-year period. At the beginning of the study and the end of each year, children who had participated in the Number Worlds program were compared with two other groups: (1) a sec- ond low-socioeconomic-status group that had originally been tested as hav- ing superior achievement in mathematics, and (2) a mixed-socioeconomic- status (largely middle-class) group that had also demonstrated a higher level of performance at the outset and attended an acclaimed magnet school with a special mathematics coordinator and an enriched mathematics program. These three groups are represented in the figure of Box 6-6, and the differ- ences between the magnet school students and the students in the low- socioeconomic-status groups can be seen in the different start positions of the lines on the graph. Over the course of this study, which extended from the beginning of kindergarten to the end of second grade, children who had taken part in the Number Worlds program caught up with, and gradually outstripped, the magnet school group on the major measure used through- out this study—the Number Knowledge test (see Box 6-6). On this measure, as well as on a variety of other mathematics tests (e.g., measures of number sense), the Number Worlds group outperformed the second low-socioeco- nomic-status group from the end of kindergarten onward. On tests of proce- dural knowledge administered at the end of first grade, they also compared very favorably with groups from China and Japan that were tested on the same measures.16 These findings provide clear evidence that a program based on the principles of How People Learn (i.e., the Number Worlds program) works for the population of children most in need of effective school-based instruc- tion—those living in poverty. In a variety of studies, the program enabled children from diverse cultural backgrounds to start their formal learning of arithmetic on an equal footing with their more-advantaged peers. It also enabled them to keep pace with their more-advantaged peers (and even outperform them on some measures) as they progressed through the first few years of formal schooling and to acquire the higher-level mathematics concepts that are central for continued progress in this area. In addition to the mathematics learning and achievement demonstrated in these studies, two other findings are worthy of note: both teachers and children who have used the Number Worlds program consistently report a positive attitude to- ward the teaching and learning of math. For teachers, this often represents a dramatic change in attitude. Math is now seen as fun, as well as useful, and both teachers and children are eager to do more of it.
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304 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM Comparing Number Worlds and Control Group Outcomes BOX 6-6 As the figure below shows, the magnet school group began kindergarten with substantially higher scores on the Number Knowledge test than those of children in the Number Worlds and control groups. The gap indicated a developmental lag that exceeded one year, and for many children in the Number Worlds group was closer to 2 years. By the end of the kindergarten year, however, the Number Worlds children had narrowed this gap to a small fraction of its initial size. By the end of the second grade, the Number Worlds children actually outperformed the magnet school group. In contrast, the initial gap between the control group and the magnet school group did not narrow over time. The control group chil- dren did make steady progress over the 3 years; however, they were never able to catch up. Number Worlds Control Magnet School Mean developmental level scores on Number Knowledge test at four time periods.
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305 FOSTERING THE DEVELOPMENT OF WHOLE-NUMBER SENSE SUMMARY AND CONCLUSION It was suggested at the beginning of this chapter that the teaching of whole-number concepts could be improved if each math teacher asked three questions on a regular basis: (1) Where am I now? (in terms of the knowl- edge children in their classrooms have available to build upon); (2) Where do I want to go? (in terms of the knowledge they want all children in their classrooms to acquire during the school year); and (3) What is the best way to get there? (in terms of the learning opportunities they will provide to enable all children in their class to reach the chosen objectives). The chal- lenges these questions pose for primary and elementary teachers who have not been exposed in their professional training to the knowledge base needed to construct good answers were also acknowledged. Exposing teachers to this knowledge base is a major goal of the present volume. In this chapter, I have attempted to show how the three learning principles that lie at the heart of this knowledge base—and that are closely linked to the three ques- tions posed above—can be used to improve the teaching and learning of whole numbers. To illustrate learning Principle 1 (eliciting and building upon student knowledge), I have drawn from the cognitive developmental literature and described the number knowledge children typically demonstrate at each age level between ages 4 and 8 when asked a series of questions on an assess- ment tool—the Number Knowledge Test—that was created to elicit this knowl- edge. To address learning Principle 2 (building learning paths and networks of knowledge), I have again used the cognitive developmental literature to identify knowledge networks that lie at the heart of number sense (and that should be taught) and to suggest learning paths that are consistent with the goals for mathematics education provided in the NCTM standards.17 To illus- trate learning Principle 3 (building resourceful, self-regulating mathematics thinkers and problem solvers), I have drawn from a mathematics program called Number Worlds that was specifically developed to teach the knowl- edge networks identified for Principle 2 and that relied heavily on the find- ings of How People Learn to achieve this goal. Other programs that have also been developed to teach number sense and to put the principles of How People Learn into action have been noted in this chapter, and teachers are encouraged to explore these resources to obtain a richer picture of how Principle 3 can be realized in mathematics classrooms. In closing, I would like to acknowledge that it is not an easy task to develop a practice that embodies the three learning principles outlined herein. Doing so requires continuous effort over a long period of time, and even when this task has been accomplished, teaching in the manner described in this chapter is hard work. Teachers can take comfort in the fact the these efforts will pay off in terms of children’s mathematics learning and achieve- ment; in the positive attitude toward mathematics that students will acquire
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306 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM and carry with them throughout their lives; and in the sense of accomplish- ment a teacher can derive from the fruits of these efforts. The well-deserved professional pride that this can engender, as well as the accomplishments of children themselves, will provide ample rewards for these efforts. ACKNOWLEDGMENTS The development of the Number Worlds program and the research that is described in this chapter were made possible by the generous support of the James S. McDonnell Foundation. The author gratefully acknowledges this support, as well as the contributions of all the teachers and children who have used the program in various stages of development, and who have helped shape its final form. NOTES 1. Referenced in Griffin and Case, 1997. 2. Griffin and Case, 1996a. 3. Ibid. 4. Gelman, 1978. 5. Starkey, 1992. 6. Siegler and Robinson, 1982. 7. Case and Griffin, 1990; Griffin et al., 1994. 8. Griffin et al., 1995. 9. Griffin et al., 1992. 10. Ball, 1993; Carpenter and Fennema, 1992; Cobb et al., 1988; Fuson, 1997; Hiebert, 1997; Lampert, 1986; Schifter and Fosnot, 1993. 11. Griffin and Case, 1996b; Griffin, 1997, 1998, 2000. 12. Schmandt-Basserat, 1978. 13. Damerow et al., 1995. 14. Griffin et al., 1994, 1995. 15. Also see Griffin et al., 1994; Griffin and Case, 1996a. 16. Griffin and Case, 1997. 17. National Council of Teachers of Mathematics, 2000. REFERENCES Ball, D.L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. Elementary School Journal, 93(4), 373-397. Carpenter, T., and Fennema, E. (1992). Cognitively guided instruction: Building on the knowledge of students and teachers. International Journal of Research in Education, 17(5), 457-470.
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307 FOSTERING THE DEVELOPMENT OF WHOLE-NUMBER SENSE Case, R., and Griffin, S. (1990). Child cognitive development: The role of central conceptual structures in the development of scientific and social thought. In E.A. Hauert (Ed.), Developmental psychology: Cognitive, perceptuo-motor, and neurological perspectives (pp. 193-230). North-Holland, The Netherlands: Elsevier. Cobb, P., Yackel, E., and Wood, T. (1988). A constructivist approach to second grade mathematics. In E. von Glasserfeld (Ed.), Constructivism in mathematics educa- tion. Dordecht, The Netherlands: D. Reidel. Dehaene, S., and Cohen, L. (1995). Towards an anatomical and functional model of number processing. Mathematical Cognition, 1, 83-120. Damerow, P., Englund, R.K., and Nissen, H.J. (1995). The first representations of number and the development of the number concept. In R. Damerow (Ed.), Abstraction and representation: Essays on the cultural evolution of thinking (pp. 275-297). Book Series: Boston studies in the philosophy of science, vol. 175. Dordrecht, The Netherlands: Kluwer Academic. Fuson, K. (1997). Snapshots across two years in the life of an urban Latino classroom. In J. Hiebert (Ed.), Making sense: Teaching and learning mathematics with un- derstanding. Portsmouth, NH: Heinemann. Gelman, R. (1978). Children’s counting: What does and does not develop. In R.S. Siegler (Ed.), Children’s thinking: What develops (pp. 213-242). Mahwah, NJ: Lawrence Erlbaum Associates. Griffin, S. (1997). Number worlds: Grade one level. Durham, NH: Number Worlds Alliance. Griffin, S. (1998). Number worlds: Grade two level. Durham, NH: Number Worlds Alliance. Griffin, S. (2000). Number worlds: Preschool level. Durham, NH: Number Worlds Alli- ance. Griffin, S. (in press). Evaluation of a program to teach number sense to children at risk for school failure. Journal for Research in Mathematics Education. Griffin, S., and Case, R. (1996a). Evaluating the breadth and depth of training effects when central conceptual structures are taught. Society for Research in Child Development Monographs, 59, 90-113. Griffin, S., and Case, R. (1996b). Number worlds: Kindergarten level. Durham, NH: Number Worlds Alliance. Griffin, S., and Case, R. (1997). Re-thinking the primary school math curriculum: An approach based on cognitive science. Issues in Education, 3(1), 1-49. Griffin, S., Case, R., and Sandieson, R. (1992). Synchrony and asynchrony in the acquisition of children’s everyday mathematical knowledge. In R. Case (Ed.), The mind’s staircase: Exploring the conceptual underpinnings of children’s thought and knowledge (pp. 75-97). Mahwah, NJ: Lawrence Erlbaum Associates. Griffin, S., Case, R., and Siegler, R. (1994). Rightstart: Providing the central concep- tual prerequisites for first formal learning of arithmetic to students at-risk for school failure. In K. McGilly (Ed.), Classroom lessons: Integrating cognitive theory and classroom practice (pp. 24-49). Cambridge, MA: Bradford Books MIT Press.
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308 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM Griffin, S., Case, R., and Capodilupo, A. (1995). Teaching for understanding: The importance of central conceptual structures in the elementary mathematics cur- riculum. In A. McKeough, I. Lupert, and A. Marini (Eds.), Teaching for transfer: Fostering generalization in learning (pp. 121-151). Mahwah, NJ: Lawrence Erlbaum Associates. Hiebert, J, (1997). Making sense: Teaching and learning mathematics with under- standing. Portsmouth, NH: Heinemann. Lampert, M. (1986). Knowing, doing, and teaching multiplication. Cognition and Instruction 3(4), 305-342. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics. Schifter, D., and Fosnot, C. (1993). Reconstructing mathematics education. New York: Teachers College Press. Schmandt-Basserat, D. (1978). The earliest precursor of writing. Scientific American, 238(June), 40-49. Siegler, R.S., and Robinson, M. (1982). The development of numerical understanding. In H.W. Reese and R. Kail (Eds.), Advances in child development and behavior. New York: Academic Press. Starkey, P. (1992). The early development of numerical reasoning. Cognition and Instruction, 43, 93-126.
Representative terms from entire chapter: