A student teacher whom I was observing in a first-grade classroom was
concluding a lesson on addition in which one of the addends was 0 and
the other, a number up to 5. After the books and materials had been put
away, she asked the children to hold up their hands if they could tell
her, without counting or using their fingers, the answer to 4 plus 0.
Most hands went up immediately, and the children agreed that the sum was
4. She then asked if anyone could predict the answer to 0 plus 9. After
a slight pause, several children raised their hands, and Josh said it
would be 9. Before the teacher could ask him to explain his answer, Emily
called out, "I know what a hundred plus zero is." Almost immediately I
heard other children yelling in succession and with great excitement,
"I can do a thousand plus zero"; "I know zero plus a million"; and "A
zillion plus zero is still a zillion!"
Algebra is sometimes defined as generalized arithmetic or as a language
for generalizing arithmetic. However, algebra is more than a set of rules
for manipulating symbols; it is a way of thinking. When the children in
the episode described realized that they could extend an idea beyond concrete
situations and small numbers, they exhibited algebraic thinking and experienced
the power of mathematics. The purpose of this article is to illustrate
how many key algebraic concepts can be informally developed within the
number-and-operations strand in the primary grades.
EXPRESSIONS
AND EQUATIONS Children first encounter expressions and equations,
often called number sentences, when they learn to record the results of
an addition situation. For example, if three ducks swimming in a pond
are joined by two other ducks, the symbolic expression 3 + 2 is used to
record the joining action and is a name for the number of ducks. The equation
3 + 2 = 5 states that 5 is another name for the total number of ducks.
The equals sign means that 3 + 2 and 5 both name the same number.
Many
children view the equals sign only as an instruction to compute. This
misconception is reinforced early on when they are shown the vertical
computational format for 3 + 2; the bar under the lower number is a signal
to find an answer. It is also true that pressing 3 + 2 = on a calculator
results in the standard form of the number.Therefore,
children need experiences in which they see and write other types of number
sentences, such as 5 = 3 + 2 and 3 + 2 = 4 + 1. To reinforce the concept
that a number can be represented by many different expressions, learners
are asked to name a given number in several different ways using two or
more numbers and one or more operations. For example, 9 can be named as
4 + 3 + 2 or 2 × 5 - 1. In a related activity, equations are to be completed
so that an expression has at least one operation on each side of the equals
sign. For example, given 7 + 5 as one side of an equation, a student might
write 7 + 5 = 2 × 6 or 7 + 5 = 10 + 5 - 3.
PROPERTIES
AND CONVENTIONS As students study the operations and learn
to compute, they encounter properties, which are inherent in the number
system, and conventions, which are socially agreed on aspects of symbolic
language. Explorations of number property provide valuable experiences
in generalizing in arithmetic. Children can discover and understand why
order does not matter in adding or multiplying two numbers. For example,
a student might justify 3 × 4 = 4 × 3 by showing how three groups of four
counters can be transformed into four groups of three and saying that
the two expressions are simply different ways of representing the same
number. Other regroupings show that 3 × 4 = 2 × 6 (two sets of six) or
3 × 4 = 12 (the standard-form answer--one 10 and two 1's). Learners apply
the commutative property when they compute 3 + 14 by counting on from
14 rather than 3, and 23 × 2 by doubling 23 rather than adding 2 twenty-three
times.
Many young learners incorrectly assume that the commutative property also
holds for subtraction and division. It is not uncommon for children to
read 2 - 5 as "5 minus 2" or to read the expression correctly as "2 minus
5" but state that the answer is 3. When the symbolic representations for
these operations are first introduced, the question "Does changing the
order of the two numbers change the answer?" needs to be raised for investigation,
if not by the students, then by the teacher. For example, children can
be asked to make up story problems corresponding to 5 - 2 and 2 - 5 and
consider whether these two expressions are equal. It is incorrect to state
that such expressions as 2 - 5 or 2÷6 "cannot be done." Before learning
about negative integers and fractions, many children have seen how a calculator
continues to count down past zero and know from experience that two pizzas
can be shared by six people. Teachers must also avoid saying that students
should "always subtract the smaller number from the bigger number" or
"divide the larger number by the smaller number."
When
students begin adding or multiplying three or more numbers, they find
that regardless of the order in which they perform the computation, the
sum or product is always the same. Children often apply the associative
property when they use thinking strategies to figure out basic facts.
For example, 8 + 5 can be thought of as (8 + 2) + 3; and 6 × 8, as 2 ×
(3 × 8). This principle is also used in a mental-arithmetic strategy for
simplifying computations, such as 7 + 4 + 6 + 3 and 57 × 25 × 4, and as
a way to check such computations as these by doing them in two different
ways.
When
learners attempt to evaluate such expressions as 7 - 5 - 2 and 3 + 2 ×
5, they find that different answers are possible depending on the order
in which the operations are performed. The issue here is communication;
it is necessary to learn the rules that other people using mathematics
have accepted and use. Parentheses and order-of-operation conventions
allow us to communicate with others. Note that many four-function calculators
are not programmed for standard order-of-operation rules but perform operations
in the order entered. With such a calculator, 6 + 2 × 3 is computed as
(6 + 2) × 3 rather than 6 + (2 × 3), where multiplication takes precedence.
Chain computation is also used in the type of classroom oral mental-arithmetic
drill in which the teacher calls out a sequence of numbers and operations,
such as 8 + 5 - 3 × 7 ÷ 10 = _____.
One
of the most important properties in arithmetic and algebra is the distributive
law of multiplication over addition. To explore this idea informally,
students with some knowledge of multiplication can be asked to find how
many objects are in the diagram in figure 1 and to do so in more than
one way. By looking at the arrangement as two groups, students can view
the number of objects as the sum of 3 × 2 and 3 × 4. From another perspective,
three rows of six objects are found. This relation can be recorded symbolically
using order-of-operation conventions: 3 × 2 + 3 × 4 = 3 × (2 + 4) = 3
× 6
The
distributive law is often applied as a strategy to figure out an unknown
multiplication fact using known facts. For example, 6 × 7 can be thought
of as 6 fives plus 6 twos or 5 sevens plus another seven. Symbolically,
6 × 7 = 6 × (5 + 2) = 6 × 5 + 6 × 2 = 30 + 12, or 6 × 7 = (5 + 1) × 7
= 5 × 7 + 1 × 7 = 35 + 7.
RELATIONSHIPS
BETWEEN OPERATIONS Another important algebraic idea involves
the inverse relationship between addition and subtraction and between
multiplication and division. Consider a set of three objects and another
set of two objects, as indicated in figure 2. Two addition and two subtraction
number sentences follow from the "combine" and "take away" interpretations
of the two operations. These two sentences, 3 + 2 = 5 and 5 - 2 = 3, record
that subtracting 2 is a way to undo the result of adding 2, and vice versa;
addition and subtraction are inverse operations.
The same relationship exists between multiplication and division. In the
array in figure 3, we interpret 3 × 4 as three groups of four; and 12
÷ 4, as finding how many groups of four are in twelve. The respective
equations confirm that dividing by 4 is a way to undo multiplying by 4.
Understanding
inverse operations and being able to recognize and write number-fact families
allow for greater flexibility in computation and prepare students to manipulate
expressions and solve equations in algebra. For example, knowing that
3 + 2 = 5 and 3 = 5 - 2 are equivalent equations can help learners understand
why x + 2 = 5 can be transformed to x = 5 - 2.
Students
should also be given opportunities and challenges to reflect on parallel
relationships among the four operations. Students working in small groups
might be asked to discuss how multiplication and division are like addition
and subtraction and how multiplication and addition are like division
and subtraction. They could then summarize their answers in writing or
draw a diagram, such as in figure 4, to show the relationships. In later
grades students should note and appreciate the connection between the
rule for subtracting an integer (add its opposite) and the rule for dividing
by a fraction (multiply by its reciprocal).
VARIABLES
The transition from arithmetic to algebra is marked by the
use of letters as mathematical objects. Variable is the concept that allows
arithmetic to be generalized. A variable is a representative for a range
of numbers. In the early grades, children encounter the notion of variable
when they find missing addends (3 + m = 5), when they verbalize number
properties (any number times zero is zero), and when they generalize number
patterns (the number of wheels is four times the number of cars). A discussion
follows of the use of variables in these three contexts--solving equations,
generalizing properties, and exploring functional relationships.
SOLVING
EQUATIONS In an equation such as 5 + n = 8, the variable n
is a placeholder for a specific unknown. The task is to solve for n--find
a number that will replace n and make the sentence true. Children are
first exposed to this idea through missing-addend problems. The variable
is commonly represented by the symbol m and is sometimes presented as
a mark that covers a hidden number, or as a frame in which to write a
missing number. The child might find the unknown number by recalling an
addition fact, by using guess and test, or by using a counting-on strategy.
GENERALIZATIONS
One classroom setting for a sentence like n + 0 = n is as a special equation
to be solved. When it is found that the statement is true for all numbers,
the equation can then be interpreted as being a symbolic way to state
that "any number plus 0 equals the same number." Coming from the other
direction, children who discover and verbalize this rule about adding
0, such as those in the opening anecdote, might then create or be shown
the equation as a symbolic way of generalizing the pattern: 4 + 0 = 4,
9 + 0 = 9, 100 + 0 = 100,...,n + 0 = n.
FUNCTIONAL
RELATIONSHIPS An expression such as 2 × n + 1 can be used as
a pattern generalizer that defines a function, one of the most fundamental
ideas in mathematics. For any value of n, the expression has a unique
value. Values of n and the corresponding values of the expression form
a set of ordered pairs (1, 3), (2, 5), (3, 7), and so on, where n = 1,
2, 3,....The relationship can also be presented in a table or a graph.
In
the primary grades, work relating to the function concept focuses on number
patterns and mathematical relationships. For example, children might explore
the problem of finding how many eyes are in a small group or in the whole
class (Howden 1989). They could use counting, pictures, or chips to model
the process and then record their findings in a table (fig. 5).
The
teacher would encourage the class to use words to describe the number
patterns and generalize the result. The children might relate the "eyes"
pattern to skip counting, counting by 2s, or adding 2 to the previous
term. The function idea is encountered when the focus is on the relationship
between the two patterns; the class is asked to predict, without continuing
the patterns, how many eyes ten people would have. In explaining the answer
of twenty eyes, a child might say that the number of eyes is equal to
the number of people added to itself, or doubled (multiplied by 2). Writing
the pattern rule as m + m or m × 2 introduces the use of a variable as
a placeholder for any number. In later grades students will learn the
convention of writing 2n to mean 2 × n.
This
function can also be expressed as an equation containing two variables,
for example, e = 2 × p or e = 2p. It is important to emphasize that the
variables e and p are interpreted as the number of eyes and number of
people rather than as abbreviations for these words. This understanding
may help children avoid in the future the well-documented error implicit
in writing the equation 6S = P rather than S = 6P to represent the statement
"There are six times as many students as professors" (Clement 1982).
As a setting for the function defined by 2n + 1, consider the problem
of making triangles using toothpicks. If the triangles are separate from
one another, three toothpicks are needed for each triangle and a pattern
rule is 3 × m. Suppose, however, that triangles share a common side. From
the chart shown in figure 6, we predict that twenty-one toothpicks are
needed to make ten triangles. The rule can be expressed as 2 × m + 1.
CONCLUSION
Emphasizing conceptual understanding, thinking processes, and mathematical
connections in the early teaching of arithmetic not only prepares children
for the formal study of algebra but makes the study of number and operations
more meaningful and intellectually stimulating. Young learners can find
patterns and regularities and generalize their experiences with numbers.
|