Meaningful Use of Symbols
Generalization Through
Exploring a Pattern
One of the most interesting and perhaps most valuable
methods of searching for generalization is to find it in the
growing physical pattern. One method of doing this is to
examine only one growth step of a physical pattern and ask
students to find a method of counting the elements without
simply counting each by one. The following problem is a
classic example of such a task, described in many resources
including Burns and McLaughlin (1990) and Boaler and
Humphreys (2005).
Activity 14.1
The Border Problem
On centimeter grid paper, have students draw an 8 × 8
square representing a swimming pool. Next, have
them shade in the surrounding squares, the tiles
around the pool (see Figure 14.3). The task is to find
a way to count the border tiles without counting them
one by one. Students should use their drawings,
words, and number sentences to show how they
counted the squares.
There are at least five different methods of counting the border tiles around a square other than counting them one at a time.
257
10 + 10 + 8 + 8 = 36 or 2 × 10 + 2 × 8 = 36
Each of the following expressions can likewise be traced
to looking at the squares in various groupings:
4×9
4×8+4
4 × 10 – 4
100 – 64
More expressions are possible, since students may use
addition instead of multiplication in the expressions. In any
case, once the generalizations are created, students need
to justify how the elements in the expression map to the
physical representation.
Another approach to the Border Problem is to have
students build a series of pools in steps, each with one more
tile on the side (3 × 3, 4 × 4, 5 × 5, etc.) and then find a way
to count the elements of each step using an algorithm that
handles the step numbers in the same manner at each step.
Students can find, for example, number sentences parallel
to what they wrote for the 8 × 8 to find a 6 × 6 pool and a 7
× 7 pool. Eventually, this can result in a generalized statement, for example, taking 2 × 10 + 2 × 8 and generalizing it
to 2 × (n + 2) + 2(n).
One important idea in generalization is recognizing a
new situation where it can apply and adapting it appropriately. For example, students may explore other perimeterrelated growing patterns, such as a triangle with 3, 4, and
5 dots on each side. Students should reason that this is the
same type of pattern, except that it has three sides, and be
able to use their previous generalization for this specific
problem (Steele, 2005).
Meaningful Use
of Symbols
Figure 14.3 How many different ways can you find to
count the border tiles of an 8 × 8 pool without counting them
one at a time?
Pause and Reflect
Before reading further, see if you can find four or five
different counting schemes for the border tiles problem. Apply
your method to a square border of other dimensions.
A very common solution is to notice that there are ten
squares across the top and also across the bottom, leaving
eight squares on either side. This might be written as:
Perhaps one reason that students are unsuccessful
in algebra is that they do not have a strong understanding
of the symbols they are using. For many adults, the word
algebra elicits memories of simplifying long equations with
the goal of finding x. These experiences of manipulating
symbols were often devoid of meaning and resulted in such
a strong dislike for mathematics that algebra has become
a favorite target of cartoonists and Hollywood writers. In
reality, symbols represent real events and should be seen as
useful tools for solving important problems that aid in decision making (e.g., calculating how many we need to sell to
make x dollars or at what rate do a given number of employees need to work to finish the project on time). Students
cannot make sense of such questions without meaningful
instruction on two very important (and poorly understood)
topics: the equal sign and variables.
258
Chapter 14 Algebraic Thinking: Generalizations, Patterns, and Functions
The Meaning of the Equal Sign
The equal sign is one of the most important symbols in
elementary arithmetic, in algebra, and in all mathematics
using numbers and operations. At the same time, research
dating from 1975 to the present indicates clearly that “=”
is a very poorly understood symbol (RAND Mathematics
Study Panel, 2003).
Pause and Reflect
In the following expression, what number do you think
belongs in the box?
8+4=
+5
How do you think students in the early grades or in middle
school typically answer this question?
In one study, no more than 10 percent of students at
any grade from 1 to 6 put the correct number (7) in the
box. The common responses were 12 and 17. (How did
students get these answers?) In grade 6, not one student out
of 145 put a 7 in the box (Falkner, Levi, & Carpenter, 1999).
Earlier studies found similar results (Behr, Erlwanger, &
Nichols, 1975; Erlwanger & Berlanger, 1983).
Where do such misconceptions come from? Most,
if not all, equations that students encounter in elementary school looks like this: 5 + 7 = ___ or 8 × 45 = ____ or
9(3 + 8) = ___. Naturally, students come to know = to signify
“and the answer is” rather than a symbol to indicate equivalence (Carpenter, Franke, & Levi, 2003; McNeil & Alibali,
2005; Molina & Ambrose, 2006).
Why is it so important that students correctly understand the equal sign? First, it is important for students
to see, understand, and symbolize the relationships in
our number system. The equal sign is a principal method
of representing these relationships. For example, 6 × 7 =
5 × 7 + 7. This is not only a fact strategy but also an application of the distributive property. The distributive property
allows us to multiply each of the parts separately: (1 + 5)
× 7 = (1 × 7) + (5 × 7). Other number properties are used to
convert this last expression to 5 × 7 + 7. When these ideas,
initially and informally developed through arithmetic, are
generalized and expressed symbolically, powerful relationships are available for working with other numbers in a
generalized manner.
A second reason is that when students fail to understand the equal sign, they typically have difficulty when it
is encountered in algebraic expressions (Knuth et al., 2006).
Even solving a simple equation such as 5x – 24 = 81 requires
students to see both sides of the equal sign as equivalent
expressions. It is not possible to “do” the left-hand side.
However, if both sides are the same, then they will remain
the same when 24 is added to each side.
Conceptualizing the Equal Sign as a Balance. Helping
students understand the idea of equivalence can be developed concretely, beginning in the elementary grades. The
next two activities illustrate how tactile objects and visualizations can reinforce the “balancing” notion of the equal
sign (ideas adapted from Mann, 2004).
Activity 14.2
Seesaw Students
Ask students to raise their arms to look like a seesaw.
Explain that you have big juicy oranges, all weighing
the same, and tiny little apples, all weighing the same.
Ask students to imagine that you have placed an
orange in each of their left hands (students should
bend to lower left side). Ask students to imagine that
you place another orange on the right side (students
level off). Next, with oranges still there, ask students
to imagine an apple added to the left. Finally, say you
are adding another apple, but tell students it is going
on the left (again). Then ask them to imagine it moving
over to the right.
After acting out the seesaw several times, ask students to write Seesaw Findings (e.g., “If you have a
balanced seesaw and add something to one side, it
will tilt to that side,” and “If you take away the same
object from both sides of the seesaw, it will still be
balanced”).
Activity 14.3
What Do You Know about the Shapes?
Present a scale with objects on both sides and ask
students what they know about the shapes. You can
create your own, but here is one as an example:
The red cylinders weigh the same. The yellow balls
weigh the same. What do you know about the weights
of the balls and the cylinders? Figure 14.4 illustrates
how one third grader explained what she knew.
(Notice that these tasks, appropriate for early grades,
are good beginnings for the more advanced balancing
tasks later in this chapter.)
After students have experiences with shapes, they can
then explore numbers, eventually going on to variables.
Meaningful Use of Symbols
259
Figure 14.5 offers examples that connect the balance to the
related equation. This two-pan-balance model also illustrates
that the expressions on each side represent a number.
Activity 14.4
Tilt or Balance
Figure 14.4 Latisha’s work on the problem.
Source: Figure 4 from Mann, R. L. (2004). “Balancing Act: The Truth
Behind the Equals Sign.” Teaching Children Mathematics, 11(2), p. 68.
Reprinted with permission. Copyright © 2004 by the National Council
of Teachers of Mathematics, www.nctm.org. All rights reserved.
(a)
Tilt!
(3 x 9) + 5
6x8
(3 × 9) + 5 < 6 × 8
Balance!
2x7
(3 x 4) + 2
(3 × 4) + 2 = 2 × 7
Tilt!
(4 + 9) x 3
5x7
5 × 7 < (4 + 9) × 3
(b)
+3
2x
Try
=5
Tilt!
5 +3<2× 5
Try
=3
3 +3=2× 3
Balance!
Figure 14.5 Using expressions and variables in equations
and inequalities. The two-pan balance helps develop the
meaning of =.
On the board or overhead, draw a simple two-pan balance. In each pan, write a numeric expression and ask
which pan will go down or whether the two will balance
(see Figure 14.5(a)). Challenge students to write expressions for each side of the scale to make it balance.
For each, write a corresponding equation to illustrate
the meaning of =. Note that when the scale “tilts,” either
a “greater than” or “less than” symbol (> or <) is used.
After a short time, add variables to the expressions
and allow students to solve them using whatever
methods they wish (see Figure 14.5(b)). Do not make
the task so easy that the solutions can be found by
simple inspection.
The balance is a concrete tool that can help students
understand that if you add or subtract a value from one side,
you must add or subtract a like value from the other side to
keep the equation balanced.
Figure 14.6 shows solutions for two equations, one in a
balance and the other without. Even after you have stopped
using the balance, it is a good idea to refer to the scale or
balance-pan concept of equality and the idea of keeping the
scales balanced.
As students begin to develop equations they wish
to graph, the equations will often be in a form in which
neither variable is isolated. For example, in the equation
3A – B = 2A, they may want A in terms of B or B in terms
of A. The same technique of solving for one variable can
be used to solve for one variable in terms of the other by
adjusting the expressions on both sides while keeping the
equation in balance.
An NCTM Illuminations applet titled Pan
Balance—Expressions provides a virtual balance where students can enter what they
believe to be equivalent expressions (with numbers or symbols) each in a separate pan to see if, in fact, the expressions
balance. ◆
True/False and Open Sentences. Carpenter, Franke,
and Levi (2003) suggest that a good starting point for helping students with the equal sign is to explore equations as
either true or false. Clarifying the meaning of the equal sign
is just one of the outcomes of this type of exploration, as
seen in the following activity.
260
Chapter 14 Algebraic Thinking: Generalizations, Patterns, and Functions
(a)
4 – 6x
3(1 + x)
Subtract 4 from both sides and multiply right-hand
expression.
–6x
3 + 3x – 4
Subtract 3x from both sides.
–9x
–1
Divide both sides by –9.
Your collection might include other operations but
keep the computations simple. The students’ task is to
decide which of the equations are true equations and
which are not. For each response they are to explain
their reasoning.
After this initial exploration of true/false sentences,
have students explore equations that are in a less
familiar form:
4+5=8+1
3+7=7+3
6–3=7–4
4+5=4+5
9 + 5 = 14
9 + 5 = 14 + 0
8=8
Do not try to explore all variations in a single lesson. Listen to the types of reasons that students are
using to justify their answers and plan additional equations accordingly for subsequent days.
1–
9
x
Check:
4 – 69–
3(1 + 19– )
1
Both sides = 3 – .
3
(b)
N
4.2N + 63 = —
2
Subtract 63.
N
4.2N = — – 63
2
Multiply by 2.
8.4N = N – 126
Subtract N.
7.4N = –126
Students will generally agree on equations where there
is an expression on one side and a single number on the
other, although initially the less familiar form of 7 = 2 + 5
may cause some discussion. For an equation with no operation (8 = 8), the discussion may be heated. Students often
believe that there must be an operation on one side. Equations with an operation on both sides of the equal sign can
elicit powerful discussions and help clear up misconceptions. Reinforce that the equal sign means “is the same as.”
Their internalization of this idea will come from the discussions and their own justifications. Inequalities should be
explored in a similar manner.
After students have experienced true/false sentences,
introduce an open sentence—one with a box to be filled
in or letter to be replaced. To develop an understanding of
open sentences, encourage students to look at the number
sentence holistically and discuss in words what the equation
represents.
Divide by 7.4. (Use a calculator!)
N = –17.03 (about)
Activity 14.6
Figure 14.6 Using a balance scale to think about solving
Open Sentences
equations.
Write several open sentences on the board. To begin
with, these can be similar to the true/false sentences
that you have been exploring.
Activity 14.5
True or False
Introduce true/false sentences or equations with simple examples to explain what is meant by a true equation and a false equation. Then put several simple
equations on the board, some true and some false.
The following are appropriate for primary grades:
5+2=7
4+1=6
4+4=8
8 = 10 – 1
5+2=
4+
=6
4+5=
3+7=7+
+4=8
6–
+5=5+8
=7–4
–1
= 10 – 1
The task is to decide what number can be put into
the box to make the sentence true. Of course, an
explanation is also required.
For grades 3 and above, include multiplication as
well as addition and subtraction.
Meaningful Use of Symbols
Initially, some students will revert to doing computations and putting the answer in the box. This is a result of
too many exercises where an answer is to be written as a
single number following an equal sign. In fact, the box is a
forerunner of a variable, not an answer holder.
Relational Thinking. Once students understand that the
equal sign means that the quantities on both sides are the
same, they can use relational thinking in solving problems.
Relational thinking takes place when a student observes
and uses numeric relationships between the two sides of
the equal sign rather than actually computing the amounts.
Relational thinking of this sort is a first step toward generalizing relationships found in arithmetic so that these same
relationships can be used when variables are involved rather
than numbers.
Consider two distinctly different explanations for placing a 5 in the box for the open sentence 7 – = 6 – 4.
a. Since 6 – 4 is 2, you need to take away from 7 to get 2.
7 – 5 is 2, so 5 goes in the box.
b. Seven is one more than the 6 on the other side. That
means that you need to take one more away on the left
side to get the same number. One more than 4 is 5 so
5 goes in the box.
261
Open Sentences
73 + 56 = 71 +
20 × 48 =
× 24
126 – 37 =
– 40
68 + 58 = 57 + 69 +
Pause and Reflect
One of the true/false statements is false. Can you
explain why using relational thinking?
Marta Molina and Rebecca Ambrose (2006), researchers in mathematics education, used the true/false and
open-ended prompts with third graders, none of whom
understood the equal sign in a relational way at the start
of their study. For example, all 13 students answered
8 + 4 = ___ + 5 with 12. They found that asking students
to write their own open sentences was particularly effective in helping students solidify their understanding of the
equal sign. The following forms were provided as guidance
(though students could use multiplication and division if
they wanted):
___ + ___ = ___ + ___
___ – ___ = ___ – ____
Pause and Reflect
How are these two correct responses actually quite different? How would each of these students solve this open
sentence? 534 + 175 = 174 +
The first student computes the result on one side and
adjusts the result on the other to make the sentence true.
The second student is using a relationship between the
expressions on either side of the equal sign. This student
does not need to compute the values on each side. When
the numbers are large, the relationship approach is much
more useful. Since 174 is one less than 175, the number
in the box must be one more than 534 to make up the difference. The first student will need to do the computation and will perhaps have difficulty finding the correct
addend.
In order to nurture relational thinking and the meaning of the equal sign, continue to explore an increasingly
complex series of true/false and open sentences with your
class. Select equations designed to elicit good thinking and
challenges rather than computation. Use large numbers
that make computation difficult (not impossible) to push
them toward relational thinking.
True/False
674 – 389 = 664 – 379
5 × 84 = 10 × 42
37 + 54 = 38 + 53
64 ÷ 14 = 32 ÷ 28
___ + ___ = ___ – ____.
Activity 14.7
Writing True/False Sentences
After students have had ample time to discuss true/
false and open sentences, ask them to make up their
own true/false sentences that they can use to challenge their classmates. Each student should write a
collection of three or four sentences with at least
one true and at least one false sentence. Encourage
them to include one “tricky” one. Their equations can
either be traded with a partner or used in full-class
discussions.
Repeat for open sentence problems.
When students write their own true/false sentences,
they often are intrigued with the idea of using large numbers and lots of numbers in their sentences. This encourages
them to create sentences involving relational thinking.
As students explore true/false and open sentence activities, look for two developments.
First, are students acquiring an appropriate understanding of the equal sign? Look to see if
they are comfortable using operations on both sides of the
262
Chapter 14 Algebraic Thinking: Generalizations, Patterns, and Functions
equal sign and can use the meaning of equal as “is the same
as” to solve open sentences.
Second, look for an emergence of relational thinking.
Students who rely on relationships found in the operations
on each side of the equal sign rather than on direct computation have moved up a step in their algebraic thinking. ◆
The Meaning of Variables
Expressions or equations with variables allow for the expression of generalizations. When students can work with
expressions involving variables without even thinking about
the specific number or numbers that the letters may stand
for, they have achieved what Kaput (1999) refers to as manipulation of opaque formalisms—they can look at and
work with the symbols themselves. Variables can be used
as unique unknown values or as quantities that vary. Unfortunately, students often think of the former and not the
latter. Experiences in elementary and middle school should
focus on building meaning for both, as delineated in the
next two sections.
Variables Used as Unknown Values. Students’ first experiences with variables tend to focus exclusively on variables as unknown values. In the open sentence explorations,
is a precursor of a variable used in this way. Early
the
on, you can begin using various letters instead of a box in
your open sentences. Rather than ask students what number goes in the box, ask what number the letter could stand
for to make the sentence true. Initial work with finding the
value of the variable that makes the sentence true—solving
the equation—should initially rely on relational thinking.
Later, students will develop specific techniques for solving
equations when these relationships are insufficient.
The balancing ideas described in the previous section
can also serve this purpose. NCTM Illuminations, for example, uses an applet titled “Pan Balance—Shapes,” along
with two excellent pre-K–2 lesson plans, for having students (virtually) weigh different shapes to figure out what
number each shape represents.
+
+7=
Consider the following open sentence:
+ 17 (or, equivalently, n + n + 7 = n + 17). A convention
for the use of multiple variables is that the same symbol
or letter in an equation stands for the same number every
place it occurs. Carpenter et al. (2003) refer to it as “the
mathematician’s rule.” In the preceding example, the
must stand for 10.
Many story problems involve a situation in which the
variable is a specific unknown, as in the following basic
example:
Gary ate five strawberries and Jeremy ate some,
too. The container of 12 was gone! How many did
Jeremy eat?
Although students can solve this problem without
using algebra, they can begin to learn about variables by
expressing it in symbols: 5 + s = 12. These problems can
grow in difficulty over time.
With a context, students can even explore three variables, each one standing for an unknown value, as in the
activity below.
Activity 14.8
Balls, Balls, Balls
How much does each ball weigh given the following
three facts:
1.
+
= 1.25 pounds
2.
+
= 1.35 pounds
3.
+
= 1.9 pounds
Ask students to look at each fact and make observations that help them generate other facts. For example,
they might notice that the soccer ball weighs 0.1
pounds more than the football. Write this in the same
fashion as the other statements. Continue until these
discoveries lead to finding the weight of each ball.
Encourage students to use models to represent and explore the problem (activity adapted from Maida, 2004).
One possible approach: Add equations 1 and 2:
+
+
+
= 2.6 pounds
Then take away the football and soccer ball, reducing
the weight by 1.9 pounds (based on the information
in equation 3), and you have two baseballs that weigh
0.7 pounds. Divide by 2, so one baseball is 0.35
pounds.
You may have recognized this last example as a system
of equations presented in a concrete manner. This type of
work is accessible to upper elementary and middle school
students when presented in this manner and helps build the
foundation for working with systems of equations later.
Another concrete way to work on systems of equations
is through balancing. Notice the work done in building the
concept of the equal sign is now applied to understanding
and solving for variables.
In Figure 14.7, a series of examples shows scale problems
in which each shape on the scales represents a different value.
Two or more scales for a single problem provide different information about the shapes or variables. Problems of this type
Meaningful Use of Symbols
can be adjusted in difficulty for children across the grades.
Greenes and Findell (1999a,b) have developed a whole collection of these and similar activities in books for grades 1 to 7.
When no numbers are involved, as in the top two examples of Figure 14.7, students can find combinations of
numbers for the shapes that make all of the balances balance. If an arbitrary value is given to one of the shapes, then
values for the other shapes can be found accordingly.
In the second example, if the sphere equals 2, then the
cylinder must be 4 and the cube equals 8. If a different value is
given to the sphere, the other shapes will change accordingly.
Pause and Reflect
How would you solve the last problem in Figure 14.7?
Can you solve it in two ways?
263
Believe it or not, you have just solved a series of simultaneous
equations, a skill generally left to a formal algebra class.
Simplifying Expressions and Equations. As noted earlier, simplifying equations and solving for x have often been
meaningless tasks, and students are unsure of what steps to
do when. Still, knowing how to simplify and recognizing
equivalent expressions are essential skills to working algebraically. In Curriculum Focal Points (NCTM, 2006), one of
the three focal points is about algebra: “Writing, interpreting, and using mathematical expressions and equations.”
Students need an understanding of how to apply mathematical properties and how to preserve equivalence as they
simplify. One way to do this is to have students look at simplifications that have errors and explain how to fix the errors
(Hawes, 2007). Figure 14.8 shows how three students have
justified the correct simplification of (2x + 1) – (x + 6).
You (and your students) can tell if you are correct by
checking your solutions with the original scale positions.
Explain how to fix this simplification. Give reasons.
(2x + 1) – (x + 6) = 2x + 1 – x + 6
Gabrielle’s solution
Which shape weighs the most? Explain.
Which shape weighs the least? Explain.
Prabdheep’s solution
What will balance 2 spheres? Explain.
8
12
How much does each shape weigh? Explain.
Briannon’s solution
13
21
14
How much does each shape weigh? Explain.
7
6
9
How much does each shape weigh? Explain.
Figure 14.7 Examples of problems with multiple variables
and multiple scales.
Figure 14.8 Three students provide different explanations for fixing the flawed simplification given.
Source: Figure 3 from Hawes, K. (2007). “Using Error Analysis to Teach Equation Solving.” Mathematics Teaching in the Middle School, 12(5), p. 241.
Reprinted with permission. Copyright © 2007 by the National Council of
Teachers of Mathematics, www.nctm.org. All rights reserved.