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From page 21... ...
In this chapter, we provide an overview of the mathematical ideas that are appropriate for preschool and the early grades and discuss some of the more complex mathematical ideas that build on them. These foundational ideas are taken for granted by many adults and are not typically examined in high school or college mathematics classes.

From page 22... ...
These process goals must be kept in mind when considering the teaching and learning of mathematics with young children. The fourth section describes connections across the content described in the first two sections as well as to important mathematics that children study later in elementary school.

From page 23... ...
Change situations: addition change plus situations (start + change gives the result) and subtraction change minus situations (start − change gives the result)

From page 24... ...
One can use numbers to give specific, detailed information about collections of things and about quantities of stuff. Initially, some toy bears in a basket may just look like "some bears," but if one knows there are seven bears in the basket, one has more detailed, precise information about the collection of bears.

From page 25... ...
It helps children solve larger problems and become more fluent in their Level 1 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 Level 2 counting of solution methods.

From page 26... ...
For example, when a child counts a set of seven bears, the child makes a 1to1 correspondence between the list 1, 2, 3, 4, 5, 6, 7 and the collection of bears. To count the bears, the child says the number word list 1, 2, 3, 4, 5, 6, 7 while pointing to one new bear for each number.

From page 27... ...
So the problem is how to give a unique name to each number. Different cultures have adopted many different solutions to this problem (e.g., Menninger, 1958/1969; see Chapter 4 of this volume for a discussion of counting words in different languages)

From page 28... ...
The 1 stands for 1 ten and the 0 stands for 0 ones, and 10 stands for the combined amount in 1 ten and 0 ones. This way of describing and writing the number ten requires thinking of it as a single group of ten  in other words, as a new entity in its own right, which Figure 22 is created by joining 10 separate things into a new coherent whole, as indicated R01420 in the figure by the way 10 dots are shown grouped to form a single unit of 10.

From page 29... ...
For example, in 37, the 3 stands for 3 tens, the 7 stands for 7 ones, and 37 stands for the combined amount in 3 tens and 7 ones. Notice that from 20 on, the way one says number words follows a regular pattern that fits with the way these numbers are written.

From page 30... ...
. If a child has a collection of black beads and another collection of white beads, and if these collections are placed near each other, the child can place each black bead with one and only one white bead.

From page 31... ...
For example, one knows that there are more beads in a col Figure 24 lection of eight black beads than there are in a collection of seven white beads because 8 occurs later inR01420 the counting list than 7 (see the bottom of Figure 24)

From page 32... ...
Viewed from a more BOX 24 Types of Addition/Subtraction Situations Change Plus and Change Minus Situations Change situations have three quantitative steps over time: start, change, result. Most children before first grade solve only problems in which the result is the unknown quantity.

From page 33... ...
In change plus and change minus situations, there is a starting quantity (A) , an amount by which this quantity changes (B)

From page 34... ...
Change plus, change minus, and put together problems in which either A or B (the start quantity, the change quantity, or one of the two parts) is unknown involve an interesting reversal between the operation that formulates the problem and the operation that can be used to solve the problem from a more advanced perspective.

From page 35... ...
could be viewed initially as an empty, unstructured whole, but objects that are placed or moved within the space begin to structure it. The beginnings of the Cartesian structure of space, a central idea in mathematics, are seen when square tiles are placed in neat arrays to form larger rectangles and when cubical blocks are stacked and layered to make larger boxshaped structures.

From page 36... ...
it takes to fill the box without any gaps. Although cubes need not be used for units of volume, they make especially useful units because they line up in neat rows and columns and stack in neat layers to fill box shapes completely without gaps.

From page 37... ...
The shapes all have an "inside region" and an "outer boundary." Distinguishing the inside region of a 2D shape from its outer boundary is an especially important foundation for understanding the distinction between the perimeter and area of a shape in later grades. Except for circles, the outer boundary of the common 2D geometric shapes consists of straight sides, and the nature of these sides and their relationships to each other are important characteristics of a shape.

From page 38... ...
Many common objects are approximate versions of these ideal, theoretical shapes. For example, a building block is a rectangular prism, and a party hat can be in the shape of a cone.

From page 39... ...
Distinguishing the inside of a 3D shape from its outer surface is an especially important foundation for understanding the distinction between the surface area and volume of a shape in later grades. Composing and Decomposing Shapes Just as 10 ones can be composed to make a single unit of 10, shapes can also be composed to make new, larger shapes.

From page 40... ...
Likewise, composing and decomposing 3D shapes is an important foundation for understanding volume in later grades. In particular, viewing rectangular prisms as composed of layers of 26 and columns of cubes is Figure rows R01420 key to understanding volumes of rectangular prisms (see Figure 27)

From page 41... ...
These two descriptions indicate relative location along two distinct (and perpendicular) horizontal axes (lines)

From page 42... ...
Note that many of the specific reasoning processes were already touched on in the discussions of number, geometry, and measurement. In fact, these specific processes represent powerful, crosscutting ideas that connect multiple concepts, procedures, or problems and can help children begin to see coherence across topics in mathematics.

From page 43... ...
and aspects of everyday life. Applying the Process Standards: Mathematizing Together, the general mathematical processes of reasoning, representing, problem solving, connecting, and communicating are mechanisms by which children can go back and forth between abstract mathematics and real situations in the world around them.

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. Specific Mathematical Reasoning Processes Mathematics learning in early childhood requires children to use several specific mathematical reasoning processes, also known as "big ideas," across domains.

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. To create repeating patterns, children must select and repeat a unit.

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Looking for Patterns and Structures and Organizing Information Looking for patterns and structures and organizing information (including classifying) are crucial mathematical processes used frequently in mathematical thinking and problem solving.

From page 47... ...
Organizing information, including classifying, has also been seen as early childhood mathematics content, as children use attribute blocks and other collections of entities in which attributes are systematically varied so that they can sort them in multiple ways. Attribute blocks usually vary in color, shape, size, and sometimes thickness, so that children can sort on any of these dimensions and also describe a given block using multiple terms.

From page 48... ...
In the realm of number, 10 individual counters are viewed as forming a single composite unit of 10. A geometric version of this grouping idea occurs when several shapes are put together to form another larger shape, which is then viewed as a unified shape in its own right, such as if the unified shape is seen as a possible substitute for another shape or as able to fill a space in a puzzle.

From page 49... ...
Groups of Groups: Numbers, Shapes,28 2D Space Figure and The compositional structure R01420 decimal system is more complex of the than just making groups of 10 from 10 ones, since every 10 groups of 10 are composed into a unit of 100. A geometric version of this group's idea occurs when shapes are put together to form a new, composite shape, and composite shapes are then put together to make another composite shape  a composite of the composite shapes.

From page 50... ...
Box shapes can be built as layers of identical cubes, as in Figure 212, and each layer can be viewed as groups of rows, so a box built from cubes can be viewed as a group of a group of cubes in the same way that 1,000 is 10 groups of 10 groups of 10. When one extends the array structure of rectangular prisms to all of 3D space, one gets essentially the idea of coordinate space, in which the location of each point in space is described by a triple of numbers that indicate its location relative to three coordinate lines.

From page 51... ...
Each property can be illustrated by moving and reorganizing objects, sometimes also by decomposing and recomposing a grouping, and sometimes even in terms of symmetry. The report Adding It Up: Helping Children Learn Mathematics has a good discussion and an illustration of the commutative and associative properties of addition, the commutative and associative properties of multiplication, and the distributive property (National Research Council, 2001, Chapter 3 and Box 31)

From page 52... ...
Recognizing this symmetry allows children to learn multiplication facts more efficiently. In other words, once they know the upper righthand triangular portion of the multiplication tables in around third grade, they can fill in the rest of the table by reflecting across the diagonal (see Figure 210)

From page 53... ...
Later in elementary school (in around second grade) , children see this inverse relationship between the size of a unit of measurement and the number of units it takes to make a given quantity reflected in the inverse relationship between the ordering of the counting numbers and the ordering of the unit fractions (see Figure 212)

From page 54... ...
. Instead of a real graph, children could display the data somewhat more abstractly in a pictograph by lining up sticky notes in categories, as on the right in the figure.

From page 55... ...
FIGURE 213 A template for a "real graph" and a pictograph made with sticky notes. Figure 213 R01420 made but not discussed are not likely to help children develop or extend their mathematical thinking.

From page 56... ...
Space has structure that derives from movement through space and from relative location within space. An important way to think about the structure of 2D and 3D space comes from viewing rectangles as composed of rows and columns of squares and viewing box shapes as composed of layers of rows and columns of cubes.

From page 57... ...
Journal for Research in Mathematics Education, 27(5)

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