How Students Learn Mathematics 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 69-120
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From page 69... ...
In this chapter, I explore each of these icebergs in turn in the context of helping children in the pnrnarv 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 pnnciples (see Chapter 1)
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From page 70... ...
Thus, when I explore question 1 (Where are we flows) 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 71... ...
) 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 72... ...
they know the counting 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")
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From page 73... ...
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 74... ...
In the first part of the kindergarten year, all the children 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)
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From page 75... ...
Because no child in the treatment 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 construction 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 76... ...
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 77... ...
(N= 11) of differences Number Knowledge Test 6-year-old level 83 100 ns 8-year-old level 0 18 Ora Arithmetic Test 33 82 Written Arithmetic Test 75 31 ns Word Problems Test 6-year-old level 54 96 8-year-old level 13 46 Teacher Rating Number sense 24 100 Number meaning 42 88 Number use 42 88 Addition 66 100 ns Subtraction 66 100 ns us= net s gn f cant; a = s on f cant at the 01 eve or better
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From page 78... ...
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-bystep 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. Aldhough dais is The subject matter of cognitive developmental psychology, it is highly relevant to teachers of young children who want to implement The developmental principles just described in their classrooms.
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From page 79... ...
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 80... ...
I'm going to show you some counting chips (Show a line of 3 red and 4 yellow chips in a row, as follows: R Y R Y R Y Y)
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From page 81... ...
Which number is closer to 21: 25 or 187 (Show visual array after asking the question.)
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From page 82... ...
Children who are unsuccessful often fail to count systematically. They say The counting words and touch The chips, but these strategies are not
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From page 83... ...
Props Needed: For Level 0: 12 red and 8 yellow counting chips, at least 1/8" thick (other contrasting colors can be substituted)
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From page 84... ...
An even less sophisticated response is given by children who have not yet learned to say the counting words in The correct sequence and who may count The four red chips in item #4 by saying, "one," "two," ~five,'' seven." By The age of 4, most children can also compare two stacks of chips dial differ in height in obvious, perceptually salient ways (Level 0, #2 and #3) and tell which pile has more or less.
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From page 85... ...
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 86... ...
The first error involves starting at seven, saying two counting words "seven, eight"—and explaining that eight is The answer The second error is to say that The answer is "eight and nine" and to repeat tills answer when prompted widh The question, "Well, which is it eight or nine? " Bodh of These answers show an understanding of The orderof counting words but a weak (or incomplete)
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From page 87... ...
Scholars hypodhesize 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 structurefor whole numher, or a mental counting line structured This structure is illustrated in Figure 6-1. The figure can be Thought of as a blueprint showing The important pieces of knowledge children have acquired (depicted by words or pictures in The figure)
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From page 88... ...
FIGURE CN1 Menhol counting line siruciure o blueprini showing ibe imporFoni pieces of knowledge children hove acquired twords or pica rest ond ibe ways Here pieces ore inierreloF d burrows] I I a lot llng4 /5 at_ |5 at_ ~ Ned ~ Ray ~ (biogrr PFn4nF .1 · W.1 111 .1 1 k~ ~ -~ T_~_ ~ ~- ~ Airmen mlmETds I 3 il 4 l s finger patterns associated with each counting word; as indicated by the horizontal and vertical arrows that connect finger displays to each other and to the counting words, they also know that the finger display contains one more finger each time they count up by one and contains one less finger each time they count down by one.
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From page 89... ...
. When this new understanding is linked to their central conceptual understanding of number, children understand dhat The numerals are symbols for number words, both as ordered "counting tags" and as indicators of set size (i.e., numerical cardinality)
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From page 90... ...
They then proceed to count the numbers in between by nodding Their heads; saying Three," "four," ~five'' (sometimes using their fingers to keep track of The second number line, in which Three" is one, "four" is two, and ~five'' is Three) ; and providing Three" as The answer This behavior suggests they are using one mental counting line as an operator to count The numbers on a second mental number line that shows The beginning and end points of The count By contrast, children who are unsuccessful with This item often give ~five" as The answer and explain This answer by saying dhat five is in between two and six Although this answer demonstrates an understanding of The order of numbers in the counting sequence, it completely ignores The part of The question that asks, "How many numbers are the e in between?
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From page 91... ...
.9 ACKNOWLEDGING TEACHERS' CONCEPTIONS AND PARTIAL I 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. My own understanding of number has grown considerably over the past several years as a result of using this test with hundreds of children, listening to what they say, and examining how their explanations and understandings change as they grow older.
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From page 92... ...
In my own work, I have found dhat 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 93... ...
REVISITING QUESTION 2: DEFINING I~IE 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 94... ...
A major goal for the kindergarten year is to ensure that children acquire a wellconsolidated 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 entails.
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From page 95... ...
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 96... ...
The Number Worlds program provides 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, At each grade level in this program, children explore f ve different lands.
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From page 97... ...
~ Additional, 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 61 ) See the Yea old level items for the prekindergarten network, the 6 year-old level items U through 6)
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From page 99... ...
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. Tally marks are used as well in this land to record and compare quantities.
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From page 100... ...
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 provides opportunities to link the "world of quantity" with the "world of counting 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. Plus Pup Plus Pup is an Object Land activity that is used in both the preschool and kindergarten programs to provide opportunities for children to (1)
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From page 101... ...
. As The icon on The card suggests, Plus Pup gives The cookie carrier one more cookie.
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From page 102... ...
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 evidence 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 103... ...
to solve Plus Pup problems and counting back (from the initial amount) to solve Minus Mouse problems will now have to employ these strategies in a much more flexible fashion.
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From page 104... ...
This experience is illustrated in The following activities. T/be Skating Party Game This game is played in Circle Land at The kindergarten level It was designed to help children realize dhat a dial (or a circular padh)
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From page 105... ...
In one variation of this game, the Award cards collected by each group of four children are computed and compared, and a group winner is declared. 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 106... ...
Let's watch next time we play and see." In encouraging children to construct their own answers to the question by reflecting on their own activity, teachers are encouraging the use of metacognitive processes and allowing children to take charge of their own learning (learning principle 3)
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From page 107... ...
, the world of counting numbers, and the world of fommal symbols. Rosetnar,u'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 108... ...
The one described in tills section possibly achieves dais purpose to a greater extent Than most odhers. It also provides 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 109... ...
As they do so, most children acquire increasingly sophisticated number competencies. For example, they become capable of performing a series of successive addition and subtraction operations in
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From page 110... ...
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 conceptual understanding are built into every Number Worlds activity, computational fluency is given special attention in the activities developed for the Warm-Up period of each lesson.
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From page 111... ...
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 padh that is f Holy attuned to Their growing understanding (learning principle 1)
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From page 112... ...
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 processes are included in the Number Worlds program.
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From page 113... ...
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 diff cult 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, 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)
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From page 114... ...
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)
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From page 115... ...
group That had also demonstrated a higher level of performance at The outset and attended an acclaimed magnet school with a special madhematics coordinator and an enriched mathematics program, These decree groups are represented in the figure of Box 6-6, and The differences between The magnet school students and The students in The lowsocioeconomic-status groups can be seen in The different start positions of the lines on the graph. Over The course of This study, which extended fiom 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 outstnpped, The magnet school group on The major measure used dhroughout This study—The Number Knowledge test (see Box 6-6)
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From page 116... ...
Number Worlds Control Magnet School Mean developmental level scores on Number Knowledge test at four time periods.
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From page 117... ...
, 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 assessment tool—the Number Knowledge Test—that was created to elicit this knowledge. To address learning Principle 2 (building learning paths and networks of knowledge)
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From page 118... ...
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
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From page 119... ...
Griffin, s., and Case, R (1996a) Evaluating the breadth and depth of t aining effects when cent al conceptual structures are taught.
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From page 120... ...
. Teaching for understanding: The importance of central conceptual st uctules in the elementary mathematics cur nculum.
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