Mathematics Learning in Early Childhood Paths Toward Excellence and Equity (2009) / Chapter Skim
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5 The Teaching-Learning Paths for Number, Relations, and Operations
5 The Teaching-Learning Paths for Number, Relations, and Operations
Pages 127-174
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From page 127... ...
Once started along these numerical learning paths, children become interested in consolidating and extending their knowledge, practicing by themselves and seeking out additional information by asking questions and giving themselves new tasks. Home, child care, and preschool and school environments need to support children in this process of becoming a self-initiating and self-guiding learner and facilitate the carrying out of such learning.
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From page 128... ...
In both the number and operations and the geometry and measurement core areas, children learn about the basic numerical or geometric concepts and objects (numbers, shapes) , and they also relate those objects and c ompose/decompose (operate on)
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From page 129... ...
This journey is full of interesting discoveries and patterns that can be supported at home and at care and education centers. THE NUMBER CORE The four mathematical aspects of the number core identified in Chapter 2 involve culturally specific ways that children learn to perceive, say, describe/discuss, and construct numbers.
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From page 130... ...
Step 2 (age 4/prekindergarten) : Use conceptual subitizing and cardinal counting of objects or fingers to solve situation, word, and oral number word problems with totals ≤ 8.
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From page 131... ...
Later 2- and 3-Year-Olds Coordinate the Number Core Components Cardinality: Continues to generalize perceptual subitizing to new configurations and extends to some instances of conceptual subitizing for 4 and 5: can give number for 1 to 5 things. Number word list: Continues to extend and may be working on the irregular teen patterns and the early decade twenty to twenty-nine, etc., pattern: says 1 to 10.
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From page 132... ...
We will call it perceptual subitizing to differentiate it from the more advanced form we discuss later for larger numbers called conceptual subitizing (see Clements, 1999)
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From page 133... ...
kinds of patterns can also be considered in terms of addends that compose them, they are included in conceptual subitizing. Such patterns can help older children learn mathematically important groups, such as five and ten; these are discussed in the later levels and in the relations and operation core discussion of addition and subtraction composing/decomposing.
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From page 134... ...
. Increasing spontaneous focusing on numerosity is an example of helping children mathematize their environment (seek out and use the mathematical information in it)
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From page 135... ...
can be counted in any arrangement, but larger sets are easier to count when they are arranged in a line. Children ages 2 and 3 who have been given opportunities to learn to count objects accurately can count objects in any arrangement up to 5 and count objects in linear arrangements up to 10 or more (Clements and Sarama, 2007; Fuson, 1988)
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From page 136... ...
Counting requires effort and continued attention, and it is normal for 4-year-olds to make some errors and for 5-year-olds to make occasional errors, especially on larger sets (of 15 or more for 4-year-olds and of 25 or more for 5-year-olds)
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From page 137... ...
For variety, these activities can involve other movements, such as marching around the room with rhythmic arm motions or stamping a foot saying a count word each time. Counting an object twice or skipping over an object are errors made occasion ally by 4-year-olds and even by 5-year-olds on larger sets.
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From page 138... ...
Coordinating the Components of the Number Core We discussed above how children coordinate their knowledge of the number word list and 1-to-1 correspondences in time and in space to count groups of objects in space. They also gradually generalize what they can count and extend their accurate counting to larger sets and to sets in various arrangements not in a row (circular, disorganized)
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From page 139... ...
So when anyone counts, they must at the end of the counting action make a mental shift from thinking of the last counted word as referring to the last counted thing to thinking of that word as referring to all of the things (the number of things in the whole set, i.e., the cardinality of the set)
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From page 140... ...
BOX 5-6 Learning the Correct Counting Language Learning the singular and plural forms that go with counting (single) and with cardinal (plural)
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From page 141... ...
Each level assumes that children have had sufficient learning experiences at the lower level to learn that content; many children can still learn the content at a level without having fully mastered the content at the lower level if they have sufficient time to learn and practice. Cardinality: Extends conceptual subitizing to 5-groups with 1, 2, 3, 4, 5 to see 6 through 10: can see the numbers 6, 7, 8, 9, 10 as 5 + 1, 5 + 2, 5 + 3, 5 + 4, 5 + 5 and can relate these to the fingers (5 on one hand)
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From page 142... ...
Knowing the 5-groups is helpful at the next level, as children add and subtract numbers 6 through 10; the patterns are problem-solving tools that can be drawn or used mentally. Children in East Asia learn and use these 5-group patterns throughout their early numerical learning (Duncan, Lee, and Fuson, 2000)
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From page 143... ...
Children in the United States tend to learn the pattern of the decade word followed by a number (1-9) before learning the order of the decade words (e.g., Fuson, 1988; Fuson, Richards, and Briars, 1982; Miller and Stigler, 1987; Siegler and Robinson, 1982)
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From page 144... ...
, with even adults not being entirely accurate. Children in kindergarten who have had adequate counting experiences earlier continue to extend their counting of objects as high as 100, often with correct correspondences (and perhaps occasional errors)
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From page 145... ...
Counting out n things also requires a conceptual advance that is the reverse of learning that the last count word tells how many there are. To count out 6 things, a child is being told how many there are (a cardinal meaning)
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From page 146... ...
1-to-1 counting correspondences: Continues to extend accurate correspon dences to larger sets; accuracy will still vary with effort: counts 25 things in a row with effort. Written number symbols: Coordinates knowledge of symbols 1 to 9 to write teen numbers: reads and writes 1 to 19; reads 1 to 100 arranged in groups of ten when counting 1 to 100.
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From page 147... ...
and so they write 81. Kindergarten children can also experience and learn all of the decade words in order from 20 to 100.
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From page 148... ...
the kindergarten concept that ten ones equal one ten can learn to use visual representations of tens that show each ten as one ten. Children at this step need to be able to make drawings of tens and of ones so that they can represent numbers to use when adding and subtracting.
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From page 149... ...
In the operations core, children learn to see addition and subtraction situations in the real world by focusing on the mathematical aspects of those situations and making a model of the situation (called mathematizing these situations, as explained in Chapter 2)
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From page 150... ...
Levels in Children's Numerical Solution Methods There is a large research base from around the world describing three levels through which children's numerical solution methods for addition and subtraction situations move (e.g., see the research summarized in Baroody, 1987, 2004; Baroody, Lai, and Mix, 2006; Clements and Sarama, 2007, 2008; Fuson, 1988, 1992a, 1992b; Ginsburg, 1983; Saxe, 1982; Sophian, 1984)
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From page 151... ...
• Use conceptual subitizing and cardinal counting to solve situation, word, oral number word, and written numeral problems with totals ≤ 10. • For word problems, model action with objects or fingers or a math drawing and count or see to solve; write an expression or equation.
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From page 152... ...
I counted on 3 more from 6 to make 9." After learning counting on from the first addend, children learn to count on from the larger addend. Count on to find the unknown addend: Children stop counting when they say the total, and the fingers (or other keeping track method)
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From page 153... ...
facilitated by children's earlier work with embedded number experiences of finding partners of a total (e.g., Inside seven, I see five and two) and by fluency with the count word sequence, so they can begin counting from any number (most 2-, 3-, and 4-year-olds need to start at 1 when counting and cannot start from just any number)
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From page 154... ...
Initially it is an action directive that means: Give me more of this. But gradually children become able to use perceptual subitizing and length or density strategies to judge which of two sets has more things: She has more than I have.
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From page 155... ...
At this step, children learn to use conceptual subitizing and cardinal counting to solve situation, word, and oral number word problems with totals ≤ 8 and begin to count and to match to find out which set has more or less (see Box 5-10)
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From page 156... ...
However, when asked to count in such situations, many 4-year-olds can count both rows accurately, remember both count words, and change them to cardinal numbers and find the order relation on the cardinal numbers (Fuson, 1988)
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From page 157... ...
Children do this in various interesting ways that can lead to productive discussions. Children also become able to use their fingers to add or to subtract using the direct modeling solution methods counting all or taking away (see Box 5-11, Level 1)
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From page 158... ...
Some children learn at home or in a care center to put the addends on separate hands, while others continue on to the next fingers for the second addend. The former method makes it easier to see the addends, and the latter method makes it easier to see the total.
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From page 159... ...
Such embedded numbers, along with the number word sequence skill of starting counting at any number, allow children to move to the second level of addition/subtraction solution procedures, counting on. Initially composed/decomposed number triads and even embedded number triads are constructed with small numbers using conceptual subitizing, but eventually counting is used with larger numbers to construct larger triads.
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From page 160... ...
It helps children solve larger problems and become more fluent in their Level 1 direct modeling solution methods. It also helps them reach fluency with the number word list in addition and subtraction situations, so that the number word list can become a representational tool for use in the counting on solution methods.
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From page 161... ...
So kindergarten children need experiences with finding and learning the partners of various numbers under 10. Children's counting and matching knowledge is now sufficient to extend to relations on sets up through 10 and to more abstract ways of presenting such relational situations as two rows of drawings that can be matched by drawing lines connecting them.
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From page 162... ...
. Change plus and change minus situations can be recorded by equations with only one number on the right because that is the action in these situations (see Box 2-4)
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From page 163... ...
Children then must shift from that last counted word to its cardinal meaning of how many objects there are in total. For example, seeing circles for both addends in a row with the problem printed above enables children to count both addends and then count all to find the total (their usual Level 1 direct modeling solution method)
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From page 164... ...
or an auditory rhythm to keep track of how many words they counted on. So here we see how the perceptual subitizing and the conceptual subitizing, which begin very early, come to be used in a more complex and advanced mathematical process.
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From page 165... ...
= 14, and students can just count on from 8 up to 14 to find that 8 plus 6 more is 14. Some first graders will also move on to Level 3 derived fact solution methods (see Box 5-11)
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From page 166... ...
They are able to use increasingly abbreviated and abstract solution methods, such as counting on and the make-a-ten methods. The number words themselves have become unitized mental objects to be added, subtracted, and ordered as their originally separate sequence, counting, and cardinal meanings become related and finally integrated over several years into a truly numerical mental number word sequence.
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From page 167... ...
The report Adding It Up: Helping Children Learn Mathematics (National
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From page 168... ...
But for numbers, relations, and operations, physical and mental number word lists are the appropriate model. Variability in Children's Solution Methods The focus of this chapter is on how children follow a learning path from age 2 to Grade 1 in learning important aspects of numbers, relations, and operations.
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From page 169... ...
SUMMARY The teaching-learning path described in this chapter shows how young children learn, integrate, and extend their knowledge about cardinality, the number word list, 1-to-1 counting correspondences, and written number symbols in successive steps from age 2 to 7. Much of this knowledge requires specific cultural knowledge -- for example, the number word list in English, counting, matching, vocabulary about relations and operations.
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From page 170... ...
. Early childhood mathematics learning.
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From page 171... ...
. Even before formal in struction, Chinese children outperform American children in mental addition.
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From page 172... ...
. Adding It Up: Helping Children Learn Mathematics.
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From page 173... ...
Journal for Research in Mathematics Education, 14, 47‑57. Shipley, E.F., and Shepperson, B
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