How Students Learn History, Mathematics, and Science in the Classroom (2005) / Chapter Skim
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6 Fostering the Development of Whole-Number Sense: Teaching Mathematics in the Primary Grades
6 Fostering the Development of Whole-Number Sense: Teaching Mathematics in the Primary Grades
Pages 257-308
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From page 257... ...
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 principles (see Chapter 1)
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From page 258... ...
) 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)
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From page 259... ...
) and provide a detailed description of the knowledge children generally have available to build upon at each age level between 4 and 8.
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From page 260... ...
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)
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From page 261... ...
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)
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From page 262... ...
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)
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From page 263... ...
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.
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From page 264... ...
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)
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From page 265... ...
(N= 11) of differencea Number Knowledge Test 6-year-old level 83 100 ns 8-year-old level 0 18 a Oral Arithmetic Test 33 82 a Written Arithmetic Test 75 91 ns Word Problems Test 6-year-old level 54 96 a 8-year-old level 13 46 a Teacher Rating Number sense 24 100 a Number meaning 42 88 a Number use 42 88 a Addition 66 100 ns Subtraction 66 100 ns ns= not significant;a= significant at the .01 level or better.
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From page 266... ...
· 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.
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From page 267... ...
It can be a great help to teachers, therefore, to have some idea of the range of understandings they can expect for children at their grade level and, equally important, to be aware of the mistakes, misunderstandings, and partial understandings 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 Number Knowledge test (see Box 6-3)
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From page 268... ...
Here are some more counting chips (show mixed array [not in a row] of 7 yellow and 8 red chips.)
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From page 269... ...
FOSTERING THE DEVELOPMENT OFWHOLE-NUMBER SENSE 269 6b. Which number is closer to 7: 4 or 9?
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From page 270... ...
The knowledge demonstrated at Level 2 represents an even more sophisti cated version of this knowledge structure. The ages associated with each level of the test represent the midpoint in the 2-year age period during which this knowledge is typically constructed and demonstrated.
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From page 271... ...
FOSTERINGTHEDEVELOPMENTOFWHOLE-NUMBER SENSE 271 BOX 6-4 Directions for Administering and Scoring the Number Knowledge Test Administration: The Number Knowledge test is an oral test. It is administered individually, and it requires an oral response.
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From page 272... ...
By the age of 4, most children have constructed an initial counting schema (i.e., a well-organized knowledge network) that en ables them to count verbally from one to five, use the one-to-one correspon dence rule, and use the cardinality rule.4 By the same age, most have also constructed an initial quantity schema that gives them an intuitive under standing of relative amount (they can compare two groups of objects that differ in size and tell which has a lot or a little)
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From page 273... ...
For the typical child this happens some time during the kindergarten year, between ages 5 and 6. With this change, children behave as if they are using a "mental counting line" inside their heads and/or their fingers to keep track of how many items they have counted.
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From page 274... ...
Two common errors that children make on this problem shed light on what suc cessful children appear to know about the number sequence. The first error involves starting at seven, saying two counting words -- "seven, eight" -- and explaining that eight is the answer.
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From page 275... ...
Again we can ask what knowledge undergirds these performances. Scholars hypothesize that, around the age of 5 to 6, as children's knowledge of counting and quantity becomes more elaborate and differentiated it also gradually becomes more integrated, eventually merging in a single knowledge network termed here as a central conceptual structure for whole number, or a mental counting line structure.7 This structure is illustrated in Figure 6-1.
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From page 276... ...
. The fifth row is connected to all the others with dotted lines to show that children acquire knowledge of the numerals that are associated with each counting word somewhat later, and this knowledge is not a vital component of the central conceptual structure.
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From page 277... ...
In effect, children realize that counting is something one can do to determine the relative value of two objects on a wide variety of dimensions (e.g., width, height, weight, musical tonality) .8 Around age 6 to 7, supported by their entry into formal schooling, children typically learn the written numerals (though this is taught to some children earlier)
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From page 278... ...
Although this answer demon strates an understanding of the order of numbers in the counting sequence, it completely ignores the part of the question that asks, "How many num bers are there in between? " Other children look stunned when this question is posed, as if it is not a meaningful thing to ask, and respond "I don't know," suggesting that they have not yet come to understand that numbers have a fixed position in the counting sequence and can themselves be counted.
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From page 279... ...
.9 ACKNOWLEDGING TEACHERS' CONCEPTIONS AND PARTIAL UNDERSTANDINGS As illustrated in the foregoing discussion, the questions included on the Number Knowledge test can provide a rich picture of the number understandings, partial understandings, and problem-solving strategies that children in several age groups bring to instruction. The test can serve another function as well, however, which is worth discussing in the present context: it can provide an opportunity for teachers to examine their own mathematical knowledge and to consider whether any of the partial understandings children demonstrate are ones they share as well.
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From page 280... ...
In my own work, I have found that the key to helping children acquire meanings for symbols is providing opportunities for them to connect the symbol system to the (more familiar) counting words.
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From page 281... ...
REVISITING QUESTION 2: DEFINING THE KNOWLEDGE THAT SHOULD BE TAUGHT Now that we have a better idea of the knowledge children have available to work with at several age levels and the manner in which this knowledge is constructed, it is possible to paint a more specific portrait of the knowledge that should be taught in school, at each grade level from preschool through second grade, to ensure that each child acquires a welldeveloped whole-number sense. As suggested previously, the knowledge taught to each child should be based, at least in part, on his or her existing
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From page 282... ...
A major goal for the kindergarten year is to ensure that children acquire a well consolidated central conceptual structure for single-digit numbers. A major goal for first grade is to help children link this structure to the formal symbol system and to construct the more elaborated knowledge network this en tails.
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From page 283... ...
To maximize opportunities for all children to achieve the knowledge objectives of the Number Worlds program, a set of design principles drawn from the How People Learn research base was adopted and used to create each of the more than 200 activities included in the program. The principles that are most relevant to the present discussion are listed below.
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From page 284... ...
The Number Worlds program pro vides one example of how these forms of representation can be taught. In so doing, it illustrates what a knowledge-centered classroom might look like in the area of elementary mathematics.
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From page 285... ...
aAdditional, more concrete, examples of the sorts of problems children can solve when they have acquired each knowledge network can be found in the Number Knowledge Test (Box 6-1)
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From page 286... ...
286 . appear they which in lands the and esentation repr of ms for five The 6-2 FIGURE
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From page 287... ...
, they gradually come to think of these patterns as forming the same sort of ordered series as do the number words themselves. Numerals, another way of representing numbers, are also part of Picture Land, and are used extensively in the activity props that are provided at all grade levels and, by the children themselves, in the upper levels of the program.
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From page 288... ...
Design Principle 2: Providing Opportunities to Link the "World of Quantity" with the "World of Counting Numbers" and the "World of Formal Symbols" Although every activity created for the Number Worlds program pro vides opportunities to link the "world of quantity" with the "world of count ing numbers" and the "world of formal symbols" -- or to link two of these worlds -- the three activities described in this section illustrate this principle nicely, at the simplest level. Readers should note that the remaining design principles are also illustrated in these examples, but to preserve the focus are not highlighted in this section.
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From page 289... ...
. As the icon on the card suggests, Plus Pup gives the cookie carrier one more cookie.
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From page 290... ...
To our delight, children who have been exposed to this activity in their preschool or kindergarten year spontaneously remember Plus Pup when they encounter more complex addition problems later on, providing evi dence they have indeed internalized the set of connections (among name, icon, and formal symbol) to which they were exposed earlier and are able to use this knowledge network to help them make sense of novel addition problems.
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From page 291... ...
Plus Pup Meets Minus Mouse Once children have become familiar with Minus Mouse and reasonably adept at solving the problems this activity presents for a range of single-digit quantities, the teacher makes the problem more complex by including both Plus Pup and Minus Mouse in the same activity. This time, when the cookie carrier walks to school, he or she draws a card from a face-down pack and either Plus Pup or Minus Mouse will surface.
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From page 292... ...
The spatial contexts that were created for the Number Worlds program often take the form of game boards on which number is depicted as a position on a line, scale, or dial and on which quantity is depicted as segments on these line, scale, and dial representations. By using a pawn to represent "self" as player and by moving through these contexts to solve problems posed by the game, children gain a vivid sense of the relationship between movement along a line, scale, or dial and increases and decreases in quantity.
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From page 293... ...
Questions are posed at several points in game play, and the sorts of questions that are put to individual children are most productive if they are finely tuned to each child's current level of understanding (learning principle 1)
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From page 294... ...
How far around the rink will you be after you do that? Is that closer to the finish line a b FIGURE 6-6 An illustrated set of skating cards used in the Circle Land Skating Party game.
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From page 295... ...
, the world of counting numbers, and the world of formal symbols. Rosemary's Magic Shoes This game provides an illustration of a spatial context developed for Line Land in the second-grade program to help children build an understanding of the base-ten number system.
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From page 296... ...
The one described in this section possi bly achieves this purpose to a greater extent than most others. It also pro vides an example of how the Number Worlds program addresses a major learning goal for first grade: helping children link their central conceptual structure for whole number to the formal symbol system.
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From page 297... ...
For this reason, this game is classified as a Picture Land activity. Children are introduced to Phase 1 of this activity by being told a story about a fire-breathing dragon that has been terrorizing the village where the children live.
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From page 298... ...
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.
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From page 299... ...
to represent the quantity depicted on the thermometer and the way this quantity changes as they count down. By systematically increasing the complexity of these activities, teachers expose children to a learning path that is finely attuned to their growing understanding (learning principle 1)
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From page 300... ...
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.
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From page 301... ...
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 Number Worlds program is a Wrap-Up period that is provided at the end of each lesson.
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From page 302... ...
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)
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From page 303... ...
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 throughout this study -- the Number Knowledge test (see Box 6-6)
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From page 304... ...
Number Worlds Control Magnet School Mean developmental level scores on Number Knowledge test at four time periods.
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From page 305... ...
and to suggest learning paths that are consistent with the goals for mathematics education provided in the NCTM standards.17 To illustrate 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 knowledge networks identified for Principle 2 and that relied heavily on the findings of How People Learn to achieve this goal.
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From page 306... ...
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.
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From page 307... ...
. Child cognitive development: The role of central conceptual structures in the development of scientific and social thought.
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From page 308... ...
. Teaching for understanding: The importance of central conceptual structures in the elementary mathematics cur riculum.
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