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In grades K-4, the
mathematics curriculum should include whole number concepts and
skills so that students can--
- construct number
meanings through real-world experiences and the use of physical
materials;
- understand our
numeration system by relating counting, grouping, and place-value
concepts;
- develop number
sense;
- interpret the multiple
uses of numbers encountered in the real world.
Focus
Children must understand
numbers if they are to make sense of the ways numbers are used in
their everyday world. They need to use numbers to quantify, to identify
location, to identify a specific object in a collection, to name,
and to measure. Furthermore, an understanding of place value is
crucial for later work with number and computation.
Intuition about number
relationships helps children make judgments about the reasonableness
of computational results and of proposed solutions to numerical
problems. Such intuition requires good number sense. Children with
good number sense (1) have well-understood number meanings, (2)
have developed multiple relationships among numbers, (3) recognize
the relative magnitudes of numbers, (4) know the relative effect
of operating on numbers, and (5) develop referents for measures
of common objects and situations in their environments.
Children come to understand
number meanings gradually. To encourage these understandings, teachers
can offer classroom experiences in which students first manipulate
physical objects and then use their own language to explain their
thinking. This active involvement in, and expression of, physical
manipulations encourages children to reflect on their actions and
to construct their own number meanings. In all situations, work
with number symbols should be meaningfully linked to concrete materials.
Emphasizing exploratory experiences with numbers that capitalize
on the natural insights of children enhances their sense of mathematical
competency, enables them to build and extend number relationships,
and helps them to develop a link between their world and the world
of mathematics.
If children are to develop
good number concepts, considerable instructional time must be devoted
to number and numeration. Children's experiences with numbers are
most beneficial when the numbers have meaning for them. A variety
of place-value tasks that assess children's thinking can be used
to identify those numbers that have meaning to individual students;
traditional numeration tasks are not good indicators of children's
understanding. Teachers can also provide exploratory experiences
with larger numbers, but symbolic tasks with numbers should not
be presented in isolation and should not be emphasized until the
numerals have been carefully linked to concrete materials and children
understand the major concepts.
Discussion
For children to use both
single-digit and multidigit number ideas fluently, written symbols
should be linked to physical models and oral names. See
figure 6.1.
Fig. 6.1
Counting skills, which
are essential for ordering and comparing numbers, are an important
component of the development of number ideas. Counting on, counting
back, and skip counting mark advances in children's development
of number ideas. However, counting is only one indicator of children's
understanding of numbers.
Understanding place value
is another critical step in the development of children's comprehension
of number concepts. Prior to formal instruction on place value,
the meanings children have for larger numbers are typically based
on counting by ones and the "one more than" relationship
between consecutive numbers. Since place-value meanings grow out
of grouping experiences, counting knowledge should be integrated
with meanings based on grouping. Children are then able to use and
make sense of procedures for comparing, ordering, rounding, and
operating with larger numbers.
The following activity
(see Fig. 6.1a) encourages children to coordinate
their counting and grouping skills to develop beginning place-value
ideas. Two children each are given the same number of counters,
in this example, thirty-two. One child counts her counters by ones;
the other groups his counters by tens and then counts by tens and
ones. The children then are asked to compare and discuss their results.
Fig. 6.1a
The next two tasks help
determine a child's place-value knowledge.
"Count these loose
chips ..... [25]. Could you write that?" [25] The teacher circles
the digit 5 and asks, "Does this part of your 25 have anything
to do with how many chips you have?" She repeats the action,
this time circling the digit 2. Children with good place-value knowledge
will match the "5" with five chips and the "2"
with twenty chips, and they may even group the twenty chips into
two groups of ten chips. [Fig. 6.2]
Fig. 6.2
"Here are 256 beans.
How many piles of 10 beans could you make?" [Fig.
6.3]
Fig. 6.3
Number sense is an intuition
about numbers that is drawn from all the varied meanings of number.
It has five components:
- Developing number
meanings. This includes the cardinal and ordinal meanings
of numbers.
- Exploring number relationships
with manipulatives. For example, the composition and decomposition
of sets of objects enables children to understand 7 as shown in
figure 6.4. Similarly, they understand
that 50 is 5 tens, 2 twenty-fives, or 4 tens and 10 ones.
Fig. 6.4
- Understanding the
relative magnitudes of numbers. For example, 31 is large compared
to 4, about the same size as 27, about half as big as 60, and
small compared to 92. Counting by ones rapidly to 100 or 1000
on a calculator helps establish the relative sizes of these numbers.
- Developing intuitions
about the relative effect of operating on numbers. This interaction
is discussed further in Standard 7, "Concepts of Whole Number
Operations," and in Standard 8, "Whole Number Computation."
- Developing referents
for measures of common objects and situations in their environment.
For example, it is unrealistic for a fourth-grade child to be
316 cm tall or to weigh 8 kg, a loaf of bread doesn't cost $117,
and the teacher is not ninety-six years old. A knowledge of reasonable
ranges for such measures provides a basis for judging reasonableness
of results.
The following classroom
example (fig. 6.5) focuses on the first two
components of number sense, the meaning of 5 and relating 5 to its
component parts.
Fig. 6.5
"Make some designs
using five toothpicks in each. Use numbers to tell me how your design
is built from the toothpicks."
The "Guess My Number"
example below helps children develop number-sense ideas regarding
the relative magnitudes of larger numbers.
The teacher tapes five
metersticks, marked in centimeters, end to end along the front of
the room. The left endpoint is labeled 0 and the right endpoint
500. One student (the selector) silently selects a number between
0 and 500, and the others try to guess it.
If the first child guesses
400 and the selector says "Too high," then that child
points to the 400 location. If the next child guesses 220 and the
selector says "Too low," then that child points to the
220 mark. Guesses continue until the secret number is guessed. The
two children pointing initially at 220 and 400 move closer together
with each guess, always bracketing the range of possibilities.
The activity in figure
6.6 focuses on relative magnitudes for even larger numbers.
Fig. 6.6
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