The concepts and skills related to number and operations are a major emphasis of mathematics instruction in prekindergarten through grade 2. Over this span, the small child who holds up two fingers in response to the question "How many is two?" grows to become the second grader who solves sophisticated problems using multidigit computation strategies. In these years, children's understanding of number develops significantly. Children come to school with rich and varied informal knowledge of number (Baroody 1992; Fuson 1988; Gelman 1994). During the early years teachers must help students strengthen their sense of number, moving from the initial development of basic counting techniques to more-sophisticated understandings of the size of numbers, number relationships, patterns, operations, and place value.
Students' work with numbers should be connected to their work with other mathematics topics. For example, computational fluency (having and using efficient and accurate methods for computing) can both enable and be enabled by students' investigations of data; a knowledge of patterns supports the development of skip-counting and algebraic thinking; and experiences with shape, space, and number help students develop estimation skills related to quantity and size.
As they work with numbers, students should develop efficient and accurate strategies
that they understand, whether they are learning the basic addition and
subtraction number combinations or computing with multidigit numbers.
They should explore numbers into the hundreds and solve problems with
a particular focus on two-digit numbers. Although good judgment must be
used about which numbers are important for students of a certain age to
work with, teachers should be careful not to underestimate what young
students can learn about number. Students are often surprisingly adept
when they encounter numbers, even large numbers, in problem contexts.
Therefore, teachers should regularly encourage students to demonstrate
and deepen their understanding of numbers and operations by solving interesting,
contextualized problems and by discussing the representations and strategies
Counting is a foundation for students' early work with number. Young children are motivated to count everything from the treats they eat to the stairs they climb, and through their repeated experience with the counting process, they learn many fundamental number concepts. They can associate number words with small collections of objects and gradually learn to count and keep track of objects in larger groups. They can establish one-to-one correspondence by moving, touching, or pointing to objects as they say the number words. They should learn that counting objects in a different order does not alter the result, and they may notice that the next whole number in the counting sequence is one more than the number just named. Children should learn that the last number named represents the last object as well as the total number of objects in the collection. They often solve addition and subtraction problems by counting concrete objects, and many children invent problem-solving » strategies based on counting strategies (Ginsburg, Klein, and Starkey 1998; Siegler 1996).
Throughout the early years, teachers should regularly give students varied opportunities to continue to develop, use, and practice counting as they quantify collections of objects, measure attributes of shapes, identify locations, and solve problems. Preschool and kindergarten teachers, for example, should use naturally occurring opportunities to help students develop number concepts by posing questions such as, How many pencils do we need at this table? Shall we count how many steps to the playground? Who is third in line? Students often use different approaches when dealing with smaller numbers versus larger numbers. They may look at a small group of objects (about six items or fewer) and recognize "how many," but they may need to count a group of ten or twelve objects to find a total. The ability to recognize at a glance small groups within a larger group supports the development of visually grouping objects as a strategy for estimating quantities.
In these early years, students develop the ability to deal with numbers mentally and to think about numbers without having a physical model (Steffe and Cobb 1988). Some students will develop this capacity before entering school, and others will acquire it during their early school years. Thus, in a first-grade class where students are asked to tell how many blocks are hidden when the total number, say, seven, is known but some, say, three, are covered, students will vary in how they deal with the covered blocks. Some may be able to note that there are four visible blocks and then count up from there, saying, "Five, six, seven. There are three hidden!" But others may not be able to answer the question unless they see all the objects; they may need to uncover and point at or touch the blocks as they count them.
As students work with numbers, they gradually develop flexibility in thinking
about numbers, which is a hallmark of number sense. Students may model
twenty-five with beans and bean sticks or with two dimes and a nickel,
or they may say that it is 2 tens and 5 ones, five more than twenty, or
halfway between twenty and thirty. Number sense develops as students understand
the size of numbers, develop multiple ways of thinking about and representing
numbers, use numbers as referents, and develop accurate perceptions about
the effects of operations on numbers (Sowder 1992). Young students can
use number sense to reason with numbers in complex ways. For example,
they may estimate the number of cubes they can hold in one hand by referring
to the number of cubes that their teacher can hold in one hand. Or if
asked whether four plus three is more or less than ten, they may recognize
that the sum is less than ten because both numbers are less than five
and five plus five makes ten.
Concrete models can help students represent numbers and develop number sense; they can also help bring meaning to students' use of written symbols and can be useful in building place-value concepts. But using materials, especially in a rote manner, does not ensure understanding. Teachers should try to uncover students' thinking as they work with concrete materials by asking questions that elicit students' thinking and reasoning. In this way, teachers can watch for students' misconceptions, such as interpreting the 2 tens and 3 ones in figure 4.1c merely as five objects. Teachers should also choose interesting tasks that engage students in mathematical thinking and reasoning, which builds their understanding of numbers and relationships among numbers. »
It is absolutely essential that students develop a solid understanding of the base-ten numeration system and place-value concepts by the end of grade 2. Students need many instructional experiences to develop their understanding of the system, including how numbers are written. They should understand, for example, that multiples of 10 provide bridges when counting (e.g., 38, 39, 40, 41) and that "10" is a special unit within the base-ten system. They should recognize that the word ten may represent a single entity (1 ten) and, at the same time, ten separate units (10 ones) and that these representations are interchangeable (Cobb and Wheatley 1988). Using concrete materials can help students learn to group and ungroup by tens. For example, such materials can help students express "23" as 23 ones (units), 1 ten and 13 ones, or 2 tens and 3 ones (see fig. 4.1). Of course, students should also note the ways in which using concrete materials to represent a number differs from using conventional notation. For example, when the numeral for the collection shown in figure 4.1 is written, the arrangement of digits mattersthe digit for the tens must be written to the left of the digit for the units. In contrast, when base-ten blocks or connecting cubes are used, the value is not affected by the arrangement of the blocks.
Technology can help students develop number sense, and it may be especially
helpful for those with special needs. For example, students who may be
uncomfortable interacting with groups or who may not be physically able
to represent numbers and display corresponding symbols can use computer
manipulatives. The computer simultaneously links the student's actions
with symbols. When the block arrangement is changed, the number displayed
is automatically changed. As with connecting cubes, students can break
computer base-ten blocks into ones or join ones together to form tens.
Place-value concepts can be developed and reinforced using calculators. For example, students can observe values displayed on a calculator and focus on which digits are changing. If students add 1 repeatedly on a calculator, they can observe that the units digit changes every time, but the tens digit changes less frequently. Through classroom conversations about such activities and patterns, teachers can help focus students' attention on important place-value ideas. Figure 4.2 shows another examplea challenging calculator activity for second gradersthat could be used to strengthen students' understanding of » place value. In this activity, students begin at one number and add or subtract to reach a target number. Since they are not limited in what they can add or subtract, activities like this allow them to use various approaches to reach the target numbers. They can decide whether to add or subtract ones or multiples of 10 or how they might use multiple steps to arrive at the target. By having students share and discuss the different strategies employed by members of a class, a teacher can highlight the ways in which students use place-value concepts in their strategies.
Students also develop understanding of place value through the strategies they
invent to compute (Fuson et al. 1997). Thus, it is not necessary to wait
for students to fully develop place-value understandings before giving
them opportunities to solve problems with two-and three-digit numbers.
When such problems arise in interesting contexts, students can often invent
ways to solve them that incorporate and deepen their understanding of
place value, especially when students have opportunities to discuss and
explain their invented strategies and approaches. Teachers emphasize place
value by asking appropriate questions and choosing problems such as finding
ten more than or ten less than a number and helping them contrast the
answers with the initial number. As a result of regular experiences with
problems that develop place-value concepts, second-grade students should
be counting into the hundreds, discovering patterns in the numeration
system related to place value, and composing (creating through different
combinations) and decomposing (breaking apart in different ways) two-and
In addition to work with whole numbers, young students should also have
some experience with simple fractions through connections to everyday
situations and meaningful problems, starting with the common fractions
expressed in the language they bring to the classroom, such as "half."
At this level, it is more important for students to recognize when things
are divided into equal parts than to focus on fraction notation. Second
graders should be able to identify three parts out of four equal parts,
or three-fourths of a folded paper that has been shaded, and to understand
that "fourths" means four equal parts of a whole. Although fractions are
not a topic for major emphasis for » pre-K2
students, informal experiences at this age will help develop a foundation
for deeper learning in higher grades.
As students in the lower grades work with complex tasks in a variety of contexts, they also build an understanding of operations on numbers. Appropriate contexts can arise through student-initiated activities, teacher-created stories, and in many other ways. As students explain their written work, solutions, and mental processes, teachers gain insight into their students' thinking. See the "Communication" section of this chapter for a more general discussion of these issues and more examples related to the development of students' understanding of number and operations.
An understanding of addition and subtraction can be generated when young students solve "joining" and take-away problems by directly modeling the situation or by using counting strategies, such as counting on or counting back (Carpenter and Moser 1984). Students develop further understandings of addition when they solve missing-addend problems that arise from stories or real situations. Further understandings of subtraction are conveyed by situations in which two collections need to be made equal or one collection needs to be made a desired size. Some problems, such as "Carlos had three cookies. María gave him some more, and now he has eight. How many did she give him?" can help students see the relationship between addition and subtraction. As they build an understanding of addition and subtraction of whole numbers, students also develop a repertoire of representations. For more discussion, see the section on "Representation" in this chapter.
In developing the meaning of operations, teachers should ensure that students
repeatedly encounter situations in which the same numbers appear in different
contexts. For example, the numbers 3, 4, and 7 may appear in problem-solving
situations that could be represented by 4 + 3, or 3 + 4, or 7 3,
or 7 4. Although different students may initially use
quite different ways of thinking to solve problems, teachers should help
students recognize that solving one kind of problem is related to solving
another kind. Recognizing the inverse relationship between addition and
subtraction can allow students to be flexible in using strategies to solve
problems. For example, suppose a student solves the problem 27 + = 36
by starting at 27 and counting up to 36, keeping track of the 9 counts.
Then, if the student is asked to solve 36 9 = ,
he may say immediately, "27." If asked how he knows, he might respond,
"Because we just did it." This student understands that 27 and 9 are numbers
in their own right, as well as two parts that make up the whole, 36. He
also understands that subtraction is the inverse of addition (Steffe and
Cobb 1988). Another student, one who does not use the relationship between
addition and subtraction, might try to solve the problem by counting back
9 units from 36, which is a much more difficult strategy to apply correctly.
In developing the meaning of addition and subtraction with whole numbers, students should also encounter the properties of operations, such as the commutativity and associativity of addition. Although some students discover and use properties of operations naturally, teachers can » bring these properties to the forefront through class discussions. For example, 6 + 9 + 4 may be easier to solve than 6 + 4 + 9, allowing students to add 6 and 4 to get 10 and 10 and 9 to get 19. Students notice that adding and subtracting the same number in a computation is equivalent to adding 0. For instance, 40 10 + 10 = 40 + 0 = 40. Some students recognize that equivalent quantities can be substituted: 8 + 7 = 8 + 2 + 5 because 7 = 2 + 5. They may realize that adding the same number (e.g., 100) to both terms of a difference (e.g., 50 10 = 40) does not change the result (150 110 = 40). The use of these properties is a sign of young students' growing number sense. Different students, however, need different amounts of time to make these properties their own. What some students learn in one year may take two or more years for others.
In prekindergarten through grade 2, students should also begin to develop an
understanding of the concepts of multiplication and division. Through
work in situations involving equal subgroups within a collection, students
can associate multiplication with the repeated joining (addition) of groups
of equal size. Similarly, they can investigate division with real objects
and through story problems, usually ones involving the distribution of
equal shares. The strategies used to solve such problemsthe repeated
joining of, and partitioning into, equal subgroupsthus become closely
associated with the meaning of multiplication and division, respectively.
Young children often initially compute by using objects and counting; however, prekindergarten through grade 2 teachers need to encourage them to shift, over time, to solving many computation problems mentally or with paper and pencil to record their thinking. Students should develop strategies for knowing basic number combinations (the single-digit addition pairs and their counterparts for subtraction) that build on their thinking about, and understanding of, numbers. Fluency with basic addition and subtraction number combinations is a goal for the pre-K2 years. By fluency we mean that students are able to compute efficiently and accurately with single-digit numbers. Teachers can help students increase their understanding and skill in single-digit addition and subtraction by providing tasks that (a) help them develop the relationships within subtraction and addition combinations and (b) elicit counting on for addition and counting up for subtraction and unknown-addend situations.
Teachers should also encourage students to share the strategies they develop
in class discussions. Students can develop and refine strategies as they
hear other students' descriptions of their thinking about number combinations.
For example, a student might compute 8 + 7 by counting on from 8: "...,
9, 10, 11, 12, 13, 14, 15." But during a class discussion of solutions
for this problem, she might hear another student's strategy, in which
he uses knowledge about 10; namely, 8 and 2 make 10, and 5 more is 15.
She may then be able to adapt and apply this strategy later when she computes
28 + 7 by saying, "28 and 2 make 30, and 5 more is 35."
Students learn basic
number combinations and develop strategies for computing that make sense
to them when they solve problems with interesting and challenging contexts.
Through class discussions, they can compare the ease of use and ease of
explanation of various strategies. In » some
cases, their strategies for computing will be close to conventional algorithms;
in other cases, they will be quite different. Many times, students' invented
approaches are based on a sound understanding of numbers and operations,
and they can often be used efficiently and accurately. Some sense of the
diversity of approaches students use can be seen in figure 4.3, which
shows the ways several students in the same second-grade class computed
25 + 37. Students 1 and 2 have represented their thinking fairly completely,
the first with words and the second with tallies. Both demonstrate an
understanding of the meaning of the numbers involved. Students 3 and 4
have each used a process that resulted in an accurate answer, but the
thinking that underlies the process is not as apparent in their recordings.
Students 5 and 6 both illustrate a common source of errortreating
the digits in ways that do not reflect their place value and thus generating
an unreasonable result.
As students work with larger numbers, their strategies for computing play an important role in linking less formal knowledge with more- » sophisticated mathematical thinking. Research provides evidence that students will rely on their own computational strategies (Cobb et al. 1991). Such inventions contribute to their mathematical development (Gravemeijer 1994; Steffe 1994). Moreover, students who used invented strategies before they learned standard algorithms demonstrated a better knowledge of base-ten concepts and could better extend their knowledge to new situations, such as finding how much of $4.00 would be left after a purchase of $1.86 (Carpenter et al. 1998, p. 9). Thus, when students compute with strategies they invent or choose because they are meaningful, their learning tends to be robustthey are able to remember and apply their knowledge. Children with specific learning disabilities can actively invent and transfer strategies if given well-designed tasks that are developmentally appropriate (Baroody 1999).
Teachers have a very important role to play in helping students develop facility with computation. By allowing students to work in ways that have meaning for them, teachers can gain insight into students' developing understanding and give them guidance. To do this well, teachers need to become familiar with the range of ways that students might think about numbers and work with them to solve problems. Consider the following hypothetical story, in which a teacher poses this problem to a class of second graders:
We have 153 students at our school. There are 273 students at the school down the street. How many students are in both schools?
As would be expected in most classrooms, the students give a variety of responses that illustrate a range of understandings. For example, Randy models the problem with bean sticks that the class has made earlier in the year, using hundreds rafts, tens sticks, and loose beans. He models the numbers and combines the bean sticks, but he is not certain how to record the results. He draws a picture of the bean sticks and labels the parts, "3 rafts," "12 tens," "6 beans" (fig. 4.4).
Ana first adds the hundreds, recording 300 as an intermediate result;
then she adds the tens, keeping the answer in her head; she then adds
the ones; and finally she adds the partial results and writes down
the answer. Her written record is shown in figure 4.5.
Some students use the conventional algorithm (stacking the addends and then adding the ones, adding the tens and renaming them as hundreds and tens, and finally adding the hundreds) accurately, but others write 3126 as their answer, demonstrating a lack of understanding that the teacher needs to address. Becky finds the answer using mental computation and writes nothing down except her answer. When asked to explain, she says, "Well, 2 hundreds and 1 hundred are 3 hundreds, and 5 tens and 5 tens are 10 tens, or another hundred, so that's 4 hundreds. There's still 2 tens left over, and 3 and 3 is 6, so it's 426."
Meaningful practice is necessary to develop fluency with basic number combinations and strategies with multidigit numbers. The example above illustrates that teachers can learn about students' understanding and at the same time get information to gauge the need for additional attention and work. Practice needs to be motivating and systematic if students are to develop computational fluency, whether mentally, with manipulative materials, or with paper and pencil. Practice can be conducted in the context of other activities, including games that require computation as part of score keeping, questions that emerge from children's literature, situations in the classroom, or focused activities that are part of another mathematical investigation. Practice should be purposeful and should focus on developing thinking strategies and a knowledge of number relationships rather than drill isolated facts.
The teacher's responsibility is to gain insight into how students are thinking about various problems by encouraging them to explain what they did with the numbers (Carpenter et al. 1989). Teachers also must decide what new tasks will challenge students and encourage them to construct strategies that are efficient and accurate and that can be generalized. Class discussions and interesting tasks help students build directly on their knowledge and skills while providing opportunities for invention, practice, and the development of deeper understanding. Students' explanations of solutions permit teachers to assess their development of number sense. As in the previous example, different levels of sophistication in understanding number relationships can be seen in second graders' responses (fig. 4.6) to the following problem. Notice that all the students used their understanding of counting by fives or of five as a unit in their solutions.
There are 93 students going to the circus. Five students can ride in each car. How many cars will be needed?
Students can learn to compute accurately and efficiently through regular experience with meaningful procedures. They benefit from instruction that blends procedural fluency and conceptual understanding (Ginsburg, Klein, and Starkey 1998; Hiebert 1999). This is true for all students, including those with special educational needs. Many children with learning disabilities can learn when they receive high-quality, conceptually oriented instruction. Special instructional interventions for those who need them often focus narrowly on skills instead of offering balanced and comprehensive instruction that uses the child's abilities to offset weaknesses and provides better long-term results (Baroody 1996). As students encounter problem situations in which computations are more cumbersome or tedious, they should be » encouraged to use calculators to aid in problem solving. In this way, even students who are slow to gain fluency with computation skills will not be deprived of worthwhile opportunities to solve complex mathematics problems and to develop and deepen their understanding of other aspects of number.
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