Does Research Tell Us about Fostering Algebraic Thinking in Arithmetic?
Algebraic thinking in arithmetic involves viewing
arithmetic with “algebra eyes” (Subramaniam & Banerjee, 2011). Cai and
Knuth (2011) call this algebraization,
and they describe it as the nature of the thinking that is basic to algebra but
also related to the conceptual areas within elementary and middle school
mathematics where such thinking can be exploited, as well as to the ways in
which teachers can help young students to develop such thinking. Blanton and
Kaput (2008) name this algebrafying
and have described it as the transforming and extending of the mathematics
normally taught in elementary school towards that of algebraic thinking with
its intrinsic feature of generality. Kieran (1996) characterizes algebraic
thinking as an approach to quantitative situations that emphasizes its general
relational aspects with tools that are not necessarily letter-symbolic.
Research (see also Carraher & Schliemann, 2007; Greenes & Rubenstein,
2008; Kieran, 2011) has begun to shape both our ways of conceptualizing
algebraic thinking and the routes by which its growth might be encouraged.
Three of its main themes are thinking relationally about equality, thinking
rule-wise in pattern generalization, and thinking representationally about the
relations in problem situations. This research brief focuses on these three
themes and describes how the teaching of arithmetic might be adapted to
encourage the development of algebraic thinking within elementary- and
Relational Meaning for the Equal Sign and for Equality
Children view the equal sign in many different ways.
Some of these ways have been found to support algebraic thinking and some not.
For example, many children, as a result of their experience in arithmetic,
erroneously think that a numerical result must always follow the equal sign
(Kieran, 1981). This first part of the research brief has two aims: (a) to describe various levels in the
ways that children view the equal sign and how these levels are related to
algebraic thinking, and (b) to
provide research-supported techniques for developing in children a relational
view of the equal sign and equality.
a study involving over 200 children from the second to the sixth grades,
Matthews et al. (2012) used an assessment instrument containing a variety of
items related to the equal sign. These items had been structured into
levels—levels that are of interest because they suggest benchmarks for the
teaching of a relational view of the equal sign:
1, the rigid operational level, involved equations with operations on the left
side of the equal sign (e.g., 4 + ◻
2, the flexible operational level, involved equations with operations on the
right side or no operations at all (e.g., Explain why “7 = 3 + 4” is true or
false; Judge “3 = 3” as true or false; 8 = 6 + ◻).
3, the basic relational level, involved equations with operations on both sides
(e.g., 8 + ◻ = 8 + 6 + 4; 10 = z + 6; Judge “7 + 6 = 6 + 6 + 1” as true
or false) and items on recognizing a relational definition of the equal sign as
being correct (i.e., “the equal sign means the same as”).
4, the comparative relational level, involved equations that could be judged
for equality by comparison methods, equations that could be solved by
compensation strategies, items on generating a relational definition of the
equal sign, and algebraic equations with more than one occurrence of the
variable (e.g., Without adding, explain why “67 + 86 = 68 + 85” is true or false;
Try to find a shortcut for “43 + ◻
= 48 + 76”; Explain without subtracting the 7 in “If 56 + 85 = 141, does 56 +
85 – 7 = 141 – 7?”; m + m + m
= m + 12, find the value of m; What does the equal sign mean?).
The researchers found
that their a priori predictions about the relative difficulty of the items were
correct. They also observed, as expected, that individual children’s ability
levels were highly correlated with grade level. Perhaps not surprisingly, the
children found that generating a relational definition of the equal sign was
more difficult than the other nonstandard equal-sign items of Levels 2 and 3.
With respect to the link between knowledge of the equal sign and algebra, two
kinds of items were especially informative to the researchers: the algebraic
equations with more than one occurrence of the variable and the advanced
relational thinking items requiring explanation of equality-preserving
with more advanced knowledge of the equal sign were more likely to solve both
types of algebraic items correctly. . . . These findings provide important
empirical evidence to support claims that young children’s knowledge of the
equal sign supports algebraic thinking. (Matthews et al., 2012, p. 339)
As this study and
others have shown, a relational view of the equal sign and of equality
constitutes a central aspect of algebraic thinking. Thus, it is important to
ask what kinds of teaching practice can assist children in cultivating such a
view. Carpenter et al. (2003) have developed a teaching approach that helps
children to see that the equal sign represents a relation between the two sides
of an equation—a relation that makes it unnecessary to calculate the total for
each side in order to determine equality. Empson et al. (2011) have used a
similar relational-thinking approach for the teaching of fractions.
The teaching sequence developed
in the Carpenter et al. study (2003, Video Case 1.5) was carried out with a
multicultural group of second graders. All of these children had answered
incorrectly the question as to the number that should go into the box of 8 + 4 =
5. Some had thought it should be 17; the others thought it should be 12. For
each of the designed tasks, the teacher would ask the children what they
thought the answer should be and listened while they justified their own point
of view or explained why they thought another student’s answer was incorrect.
The first set of tasks included only examples that the children were familiar
with, all having a single operation on the left side and for which they were
asked whether the equations were true or false (e.g., 3 + 5 = 8, 2 + 3 = 7, 58
+ 123 = 115). A few days later, the set shown below was presented to the
children. Notice the sequencing in these tasks—from a standard equation, to one
with 0 added to the right-hand element, to one where the order of these two
right-hand elements was inverted, to one where it was not the total that was
featured on the right side but rather the same decomposition as was featured on
the left side, followed lastly by a non-equality where the sum of the left-hand
member was the first numerical term of the right side.
+ 6 = 8
+ 6 = 8 + 0
+ 6 = 0 + 8
+ 6 = 2 + 6
+ 6 = 6 + 2
+ 6 = 8 + 1
The next sequence of
tasks included open-number sentences for which the children were to suggest
which number should go into the box (e.g., 3 + 5 = 3 + ◻). This was followed by examples
such as 5 + 3 = ◻ + 2. By
now, the children were gradually becoming fluent in comparing both sides and
doing mental movements of numbers. While the tasks themselves were extremely
important to the growth of these children’s understanding of equality and the
use of the equal sign, so too was the way in which the teacher encouraged them
to express their thinking and to try to justify it. In summary, the algebraic
reasoning that was fostered by the tasks of the Carpenter et al. (2003) study
allowed the children to think about arithmetic operations as relations between
numbers rather than as computational problems and to compare equalities with
several terms on each side by decomposing or rearranging some of the number
combinations, thereby demonstrating the numerical equilibrium of the equality.
The children thus came to view number sentences in a manner that prefigured
their later work with algebraic equations.
Rules within Pattern Generalization
Pattern generalization is widely considered to be an
activity suitable for developing algebraic thinking. Moreover, it is similarly
accepted that this type of activity does not require the use of literal symbols
in order to be viewed as algebraic—the core of algebraic activity being
rather a matter of thinking in certain ways. Furthermore, Radford (2011) has
argued that it is not in the generalizing
aspect per se that an activity is algebraic, for generalizing is not specific
to algebra; it is a trait of both humans and certain animals. Rather the
activity of pattern generalization is algebraic when it involves generating rules to calculate with, that is, when
children’s thinking shifts from the purely numeric to the devising of
calculation methods. However, we are not talking here about calculation methods
based on a recursive approach, or growing
rules as they are sometimes called (Warren & Cooper, 2008), but rather about
rule-based calculation methods that are sometimes referred to as position rules.
(2011) provides us with an example of such algebraic thinking in a study
carried out with second graders. The children, who worked in small groups, were
presented with the first four figures of a sequence of square-shapes (see fig.
1). They were initially asked to draw figures 5 and 6 of the sequence. As
Radford notes, the children had to first notice a commonality and then generalize
it to other terms of the sequence. But in being asked shortly thereafter to
find the number of squares for “big” figures (such as figures 25 and 50 of the
exercise), they were encouraged to go beyond the limits of their calculating
capabilities. But this movement to big figures was done in a special way.
finding the actual number of squares in the “big” figures (with the help of a
calculator), the teacher asked the children to come up with an idea of how to
find the total. This was an important strategic choice on her part because, by
going beyond the children’s arithmetic thinking abilities (i.e., counting), the
activity could shift to calculation methods or rules. Then, the children were
asked the following question dealing with a “big” unspecified figure: “Pierre
wants to build a big figure of the sequence. Explain to him what to do.”
One student, Carl, suggested, “How
about doing 500 plus 500?” to which another child in the group responded, “No;
do something simpler.” But Carl continued: “Whatever—500 plus 500 equals 1000,
plus 1, equals 1001.” Note Carl’s use
of the term, “whatever”—that is, it did not matter which “big” figure was
chosen because the underlying rule was the same. Radford insists that these
children’s generalizations were algebraic because they dealt with indeterminate
quantities and were conceived of analytically, as if they were known and could
be calculated with. The children could deal with any particular figure of this
sequence regardless of its size. The rule they were using was exemplified by a
particular case (for example, “12 plus 12, plus 1”), but it was a case where
the numbers were being used to express a more general rule: “a number plus
itself, plus 1.” The children were, in fact, using a kind of generic example. A
particular case may appear to be the focus, but it is not actually used as a
particular case but rather as an example of a general class. This generic kind
of non-letter-symbolic, verbal expression of the rules underlying patterns is
central to children’s engaging in algebraic thinking.
Other research with
slightly older children and somewhat more difficult pattern generalization
tasks has also emphasized the importance of the general rule in children’s work
with patterns. One of the pattern problems given by Moss et al. (2008) to fourth
graders involved an arrangement of trapezoid tables and chairs—the Lunchroom
Table problem (see fig. 2). The children were asked to find a rule to predict
how many chairs (dots) would fit around 56 trapezoid tables when the tables
were positioned as indicated (e.g., a rule such as “multiply the number of
tables by 3 and then add 2 for the end tables”).
One of the
difficulties that children have with generating the rule underlying patterns becomes
particularly evident when a table of values (sometimes called a T-table) is
used for recording values. Children tend to use recursive strategies (such as “add
3 for each new table”) that block them from seeing an underlying generic rule.
Such a rule would allow them to calculate, in this case, the total number of
chairs for any number of trapezoidal tables in the given arrangement. In
earlier research, Moss (2005) suggested that tables of values be used only
sparingly, if at all, in such patterning tasks with young children because of
their seductive power to induce recursion-based rules.
In the Moss et al.
(2008) study, before being presented with the Lunchroom Table problem, the
children worked with geometric growing patterns using position cards and blocks.
The position cards helped the children to focus on the relation between the
position number of an arrangement and the number of blocks used in that
particular arrangement. By asking for “far” predictions, the researchers were
able to encourage the children to think about more general functional rules
underlying the patterns rather than locally based recursive rules.
Cai and Moyer (2008)
have pointed out that, in past international studies, “U.S. sixth-grade
students tended to use concrete, arithmetic problem-solving strategies” for
problems such as the Odd-Number Pattern problem (see fig. 3). Students noticed
that each time the doorbell rang two more guests entered than on the previous
ring, so they made a table or a list by adding enough 2s sequentially in order
to answer the question about the number of rings that had occurred when 99
guests entered (Question C).
The use of such
concrete arithmetic strategies led to a much lower success rate than was the
case for students from other countries who used algebraic reasoning strategies
and who had deduced that the number of guests entering on a particular ring
was, say, “2 times the ring number minus 1.” The evidence from the studies
carried out with second and fourth graders, which has been presented above,
makes clear that the development of general computational rules in the context
of pattern generalization activity by elementary school children is a reachable
the Relations in Word Problems by Pictorial Equations
Elementary school children use a variety of
representations to signify the relations in the mathematical problems they are
asked to solve—some of these representations providing a potential bridge
between arithmetic and algebra (see, e.g., Russell et al., 2011). One
representation that has been found to be especially conducive to developing
algebraic thinking (because it prefigures the use of unknowns and algebraic
equations in the solving of word problems) is the so-called pictorial-equation
method (e.g., Cai et al., 2011; Ferruci et al., 2008).
To illustrate, we will
use as an example the Sharing Problem (see fig. 4), drawn from Ng (2004). A
central feature of the pictorial-equation method is that unknowns are
represented by unit rectangles. In the Sharing Problem example, Samy’s
rectangle or unit is the generator of all the relationships presented in the
problem. Raju, who has $100 more than Samy, is represented by a unit that is
identical to Samy’s, plus another different-sized rectangle representing the
relational portion of $100 more. A pictorial equation is formed that involves
the two basic units, the $100 rectangle, and the total amount of $410. The
entire structure of the drawing is the pictorial equation. The corresponding
literal-symbolic formulation, for higher-grade students, would involve
representing each of the units by the unknown x, within the algebraic equation x + x + 100 = 410.
students do not usually learn syntactic methods of algebraic equation solving
until about age 13 or later, the pictorial equation illustrated in figure 4 is
solved by first undoing the relational part of $100: 2 units = 410 – 100, which
works out to 310. Next, the value of one unit is calculated by carrying out a
second undoing process: if 2 units are equivalent to 310, then to find one unit
it is necessary to divide 310 by 2. (Note that the frame that includes the
problem statement in Fig. 4 also includes four additional query lines related
to the units and their values; these lines would not be filled in until the
problem is being worked out.)
carried out by Ferrucci et al. (2008) show that the pictorial approach, or model method as they refer to it, is a
valuable tool for elementary school teachers in their efforts to “enhance
mathematical problem-solving outcomes for students at multiple achievement
levels” (p. 208). Ferrucci et al. provide several examples of the ways in which
this approach can be used, including one involving the following Basketball
Game problem with fractions: “3/7 of the children at a basketball game are
girls. 48 of the children are boys. How many girls are at the basketball game?”
To generate the
pictorial equation for the Basketball Game problem, students are encouraged to
draw seven contiguous unit rectangles (indicating the fractional units), to
represent all the children at the game, and then to mark off 3 out of the 7
rectangles to represent the girls (see fig. 5). The other four rectangles,
which represent the boys, are labeled with the numerical value of 48. Students
can then calculate the value of one of these rectangles (48 ÷ 4) and conclude that
the three rectangles that represent the girls have a total value of 36 (i.e.,
12 × 3). This approach parallels that
of using the algebraic equation 4x = 48,
where x represents 1/7 of the
children at the basketball game, and where multiplying the solved-for value of x by 3 gives the number of girls.
As has been pointed
out, the pictorial-equation method offers an algebra-like approach for
representing relationships in word-problem situations—one that is well suited
to the upper-elementary and middle school student. These pictorial equations
have also been found to provide a smooth transition to the more abstract
representations of algebraic equations with their literal-symbolic notation
that are encountered in later grades.
This research brief
has described three ways in which the mathematics normally taught in elementary
and middle school can be transformed and extended towards that of algebraic
thinking. Examples included thinking relationally rather than operationally
about equality and the equal sign, thinking about general rules that allow for
calculating any single object in pattern sequences, and thinking deeply about
the relationships in word-problem situations and representing these
relationships with pictorial equations. The research presented in this brief
has offered suggestions by which teachers can help students to develop such
By Carolyn Kieran
Québec à Montréal, Canada, (Emerita)
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