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Algebraic Thinking in Arithmetic Research Brief

What Does Research Tell Us about Fostering Algebraic Thinking in Arithmetic? 

(PDF)  

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 middle-school-aged children.

Constructing 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.

In 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:

  • Level 1, the rigid operational level, involved equations with operations on the left side of the equal sign (e.g., 4 + ◻ = 8).
  • Level 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 + ).
  • Level 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”).
  • Level 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 transformations:

Children 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.

2 + 6 = 8

2 + 6 = 8 + 0

2 + 6 = 0 + 8

2 + 6 = 2 + 6

2 + 6 = 6 + 2

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.

Generating 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.

Radford (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.

 First four figures of a sequence of square-shapes 

Before 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”).

The lunchroom table problem 

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 odd-number pattern problem

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 algebraic-reasoning goal.

Representing 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.

The sharing problem and its pictorial equation

Because 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.)

Studies 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.

 A pictorial equation for the basketball game problem 

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 thinking.

By Carolyn Kieran 
Université du Québec à Montréal, Canada, (Emerita)
Michael Fish, Series Editor

References 

Blanton, Maria L., and James J. Kaput. (2008). Building district capacity for teacher development in algebraic reasoning. In James J. Kaput, David W. Carraher, and Maria L. Blanton (Eds.), Algebra in the early grades (pp. 361–388). New York: Routledge.

Cai, Jinfa, and Eric Knuth. (Eds.). (2011). Early algebraization. New York: Springer.

Cai, Jinfa, and John Moyer. (2008). Developing algebraic thinking in earlier grades: Some insights from international comparative studies. In Carole Greenes and Rheta Rubenstein (Eds.), Algebra and algebraic thinking in school mathematics (70th Yearbook of the National Council of Teachers of Mathematics, pp. 169–180). Reston, VA: NCTM.

Cai, Jinfa, Swee Fong Ng, and John C. Moyer. (2011). Developing students’ algebraic thinking in earlier grades: Lessons from China and Singapore. In Jinfa Cai and Eric Knuth (Eds.), Early algebraization (pp. 25–41). New York: Springer.

Carpenter, Thomas P., Megan Loef Franke, and Linda Levi. (2003). Thinking mathematically: Integrating arithmetic and algebra in elementary school (with accompanying video cases). Portsmouth, NH: Heinemann.

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Kieran, Carolyn. (2011). Overall commentary on early algebraization: Perspectives for research and teaching. In Jinfa Cai and Eric Knuth (Eds.), Early algebraization (pp. 579–593). New York: Springer.

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