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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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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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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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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.