Rational Errors and the Mathematical Mind

Review of General Psychology 1998. Vol. 2, No. 4,366-383

Copyright 1998 by the Educational Publishing Foundation 1089-2680/98/S3.00

Rational Errors and the Mathematical Mind Talia Ben-Zeev Brown University What do errors reveal about the mathematical mind? Intriguingly, errors are often logically consistent and rule based rather than being random. Investigating errors, therefore, presents an opportunity for uncovering the mental representations underlying mathematical reasoning. A useful question is whether errors break down into different categories or types. If this were the case, then one could explain a variety of seemingly different problem-solving behaviors by using only a few principles. The aim of this article is to provide a taxonomy of rule-based errors in mathematical reasoning that illustrates how a few basic mental processes may be responsible for generating myriad different errors. Implications for general processes of reasoning and problem solving are discussed.

(VanLehn, 1986):

Human problem solving is paradoxical. On the one hand, when people are faced with solving an unfamiliar problem, they usually do not give up but construct rules or strategies in order to solve it (Ashlock, 1976; J. S. Brown & VanLehn, 1980; Buswell, 1926; Cox, 1975; Lankford, 1972; VanLehn, 1983). These strategies tend to be systematic and internally consistent rather than being random, and often these strategies make "sense" to the people who created them. On the other hand, these same procedures often lead to erroneous solutions. What goes wrong in the problem-solving process? At what point does a logically consistent algorithm result in an error? What can these rule-based errors or rational errors (Ben-Zeev, 1995,1996) reveal about processes of reasoning and problem solving? In this article, I intend to demonstrate how errors that are rule based, deliberate, and systematic open a window into the mathematical mind by pointing to principled (misunderstandings. For instance, consider the commonly observed subtraction error illustrated below

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I wish to thank, in alphabetical order, Henry Kaufman, John Kihlstrom, Liz Phelps, Bob Sternberg, and Mike Tarr for their extremely valuable and insightful comments on earlier versions of this article. The writing of this article was made possible by a faculty grant from Brown University. Correspondence concerning this article should be addressed to Talia. Ben-Zeev, Department of Cognitive and Linguistic Sciences, Brown University, Box 1978, Providence, Rhode Island 02912. Electronic mail may be sent to Talia [email protected].

What does this error reveal about the logic of the student who produced it? VanLehn (1986) argued that this error stems from applying a smaller-from-larger rule. That is, when a student does not know how to borrow, he or she will subtract the smaller digit (e.g., the 3) from the larger digit (e.g., the 7) regardless of the position of each. This error can be considered a rational error because the student's behavior makes probabilistic sense. In past problem-solving episodes, namely, single-digit subtraction, the student has always learned to subtract smaller from larger numbers (negative numbers are not taught at this point in the curriculum). Thus, the student applies a rule that has worked in a past problem-solving episode successfully to a new and similar episode.1 A useful question is whether there are broad categories of rational errors. If so, could we explain a variety of seemingly different behaviors by using only a few principles? To date, there has not been such an integrative account of 1 Rational errors do not include "fact errors," such as 3 x 4 = 7, that result from associations or priming effects (e.g., an intrusion of an addition fact in a multiplication problem; Zbrodoff & Logan, 1986). In contrast to fact errors, rational errors refer to a higher level problem-solving performance that goes beyond simple memory effects.

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error production in mathematical cognition. the next section provides a phenomenology of There is thus a need for (a) a comprehensive rational errors. review of the literature on rational errors and (b) a categorization scheme or a taxonomy that The Nature of Rational Errors would set the stage for future empirical and in Mathematical Thinking: A Few computational investigations. Examples From Counting to Calculus The aim of this article is to provide a first step toward satisfying these goals by introducing a This section provides a description of comnew taxonomy of rational errors in mathemati- mon rational errors and highlights the rule-based cal reasoning. The taxonomy is built on processes that may govern their production. domain-specific knowledge in mathematics but has implications toward more general theories of human cognition also. For example, in Counting language development, young children have In the process of learning to count, children been shown to make grammatical errors on quickly learn how to add on single digits to a verbs they had previously acquired correctly, two-digit number (e.g., "Twenty-one, twentysuch as erroneously adding the past form -ed, to two, twenty-three . . . twenty-nine"), but once roots of familiar verbs (e.g., when children say they add on the "nine" and are asked to proceed goed when previously they said went; see to the next two-digit number, children often Karmiloff-Smith, 1986). This process of overgen- experience difficulties (Ginsburg, 1996). In eralization or induction is elaborated on in the order to solve the problem, many children discussion of the new taxonomy presented in decide that after twenty-nine, for example, this article. comes twenty-ten. This error is rational because In fact, there are many examples of system- although twenty-ten is not the right linguistic atic errors in human cognition, such as attribu- label for thirty, it nevertheless captures its tion biases in social psychology (e.g., Jones & magnitude correctly (for a more detailed acDavis, 1965; Jones & Nisbett, 1971, 1987; L. count of children's rule-based inventions in Ross, 1977), biases in reasoning under uncer- counting, see German & Meek, 1986; Siegler & tainty in cognitive psychology (e.g., Kahneman Shrager, 1984). & Tversky, 1973), and the cognitive distortions found in the clinical literature (e.g., Beck, 1967, Algebraic Problem Solving 1985, 1991) among populations such as the clinically depressed, to name just a few. This In the domain of algebraic manipulation, research on human error follows a historical students commonly produce the following tradition. Piaget, for example, was as interested precedence error (Payne & Squibb, 1990; in children's erroneous answers to test problems Sleeman, 1984): mX + n=>(n + m)X, where => as he was in their correct ones, for understand- stands for erroneous equivalence, m and n are ing human intelligence. numbers, and X is the unknown. Payne and The new taxonomy presented in this article, Squibb (1990) argued that the precedence error therefore, should be of interest to researchers in may result, in part, from forming an analogy to a variety of fields, including but not confined to language constructions such as three apples plus mathematical cognition, language, reasoning, four gives seven apples. and problem solving. The taxonomy may also prove useful for educators. One of its educa- Geometrical Problem Solving tional implications is that instead of targeting a myriad of different errors, the teacher could Dugdale (1993) described an interesting error focus on correcting the few principles that may that occurred in a high school geometry class in underlie them (also see Hennessy, 1993). which students were asked to match polynomiBefore presenting the specifics of the tax- als to functions (i.e., they were shown a graph onomy, the reader is invited to get a flavor of a and asked to write its corresponding equation). few common rational errors from different The error consisted of confusing the >• intercept mathematical problem-solving domains. Thus, of a parabola with its vertex (i.e., the "visual"

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center of the graph). For example, students who commit this error would decide that the v intercept in Figure 1A is —1 instead of —.6. Dugdale explains the confusion between the y intercept and the vertex by pointing out that in previous examples students were given, the v intercept had always coincided with the vertex of the parabola (see Figure 1B and Figure 1C). The students had thus invented a functional invariance between the two features. This error illustrates that students may pick up on spurious correlations that the teacher had not intended them to.

Calculus Several researchers have examined students' understanding of the concept of limit in calculus (e.g., Davis & Vinner, 1986; Dreyfus, 1990). They found that a common error was erroneously to assert that the terms in an infinite sequence get closer and closer to the limit but never reach it, such that for all n,an^ L (where an is the nth term of the sequence and L is the limit value). This conception may be rational because it applies to the most frequently encountered examples of infinite sequences, namely, monotonic sequences such as, .1, .01, .001, .0001 It fails, however, because it does not hold for all sequences, such as 1, 1, 1, 1,.. ., where an — L.

There is no single scheme that helps explain the origin of the above errors. In fact, error production has not been subject to much empirical research and has been dominated by primarily observational and computational investigations. To date, the main contribution to understanding errors in mathematical cognition comes from J. S. Brown and VanLehn's (1980) repair theory and its later version, Sierra (VanLehn, 1987, 1990). Even though these theories provide an invaluable account of error production, they only cover a small subset of errors, and thus only convey a part of the rational error story. As I demonstrate shortly, there is a need for a more general taxonomy of errors. The Origin of Rational Errors: Contributions and Limitations of Repair Theory and Sierra Repair theory and Sierra are examples of a computational approach to testing hypotheses about procedural learning by implementing people's problem-solving behavior in computer algorithms (Anderson, 1993; J. S. Brown & VanLehn, 1980; Langley & Ohllson, 1984; VanLehn, 1983, 1986; Young & O'Shea, 1981). The computational modeling of procedural errors is most often a variation on a production system, made up of ffC then A rules, where C is

Figure 1. The confusion between the vertex and y intercept. In Panel A, the parabola is nonsymmetrical about the y axis, resulting in different values for the vertex and the y intercept. In Panel B, the parabola is symmetrical about the y axis, resulting in the same positive value for (he vertex and the y intercept. In Panel C, the parabola is symmetrical about the y axis and is at the origin, resulting in a value of 0 for both the vertex and the y intercept.

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the condition and A is the action (see Klahr, Langley, & Neches, 1987). For instance, when C corresponds to a situation where the top digit is smaller than the bottom digit in a multicolumn subtraction problem, then A may correspond to a borrowing action. Young and O'Shea (1981) created a production-system model for simulating children's subtraction errors. Their first step was to use condition-action rules to model correct subtraction. Then, they added or omitted rules from the correct production system to simulate how students produce rational errors. For example, Young and O'Shea modeled the subtract-thesmaller-from-larger error, presented at the beginning of this article, by omitting a rule that triggers the borrowing procedure. Even though Young and O'Shea's productionsystem approach is useful for constructing a fine-grained analysis of error production, it suffers from being post hoc because thenproduction system was designed to match the student's actions after the student has already executed them. Young and O'Shea did not provide a theoretical motivation for why particular rules are added to or deleted from the correct subtraction algorithm in order to simulate the student's production of rational errors. In contrast, J. S. Brown and VanLehn's (1980) repair theory, and its later development into Sierra (VanLehn, 1983,1990), is a theory-driven computational model of rational errors. Brown and VanLehn suggest that a child first learns to execute a prefix of the subtraction algorithm. The prefix incorporates the current instructional segment of the subtraction curriculum (e.g., how to borrow from a non-zero digit) and all segments leading up to it (e.g., how to subtract without borrowing). The student's mental representation of the prefix is called a core procedure. When the child attempts to solve a problem that requires adding new rules to his or her core procedure, the student reaches an impasse. The student then selects a repair, or a set of actions that modifies the core procedure and gets the student "unstuck." Repair theory (J. S. Brown & VanLehn, 1980), however, leaves open the question of how people select the particular solution strategies that get them unstuck in the first place. More recently, VanLehn (1987,1990) attempted to answer this question by forming a computational theory called Sierra. In Sierra's framework, solution strategies are induced from

examples. For instance, a student who has only learned how to borrow on two-column subtraction problems may induce that borrowing only occurs in the adjacent and left-most digit because in all the examples the student receives, borrowing only occurs in this digit. This overgeneralization leads to errors on problems that have more than two columns such as the following (VanLehn, 1986):

-219 312 The student who demonstrates this error has adequately borrowed a 10 in the units column (11 - 9 = 2), but instead of decrementing the top digit in the ten's column, the student decided to decrement the left-most top digit in the hundred's column. As this example illustrates, the nature of this type of induction relies on a syntactic process of manipulating symbols without regard to the procedure's underlying principles. This kind of problem solving can therefore be termed as syntactic induction. Sierra's account is extremely valuable because it allows for an a priori prediction of errors. That is, Sierra predicts what kind of mistakes people would make before people actually commit them. However, Sierra suffers from a few limitations that prevent it from being a general theory of rational errors. Specifically, it focuses on a particular kind of induction (i.e., syntactic); encompasses repairs that are only preceded by impasses and therefore only occur during the execution of a procedure; and focuses on subtraction errors, which are clearly a small subset of all mathematical errors (Davis, 1982). Each of these points is discussed in more detail in the following subsections.

There Is More and Less to Rational Errors Than Syntactic Induction Although syntactic induction is an important mechanism of error production, it does not capture the full nature of rational errors in mathematical reasoning. For example, errors may occur from semantic induction, or problem solvers' active attempts to understand the problem situation and map it to everyday life experience. For instance, many errors occur on

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problems that involve the digit 0 because students tend to think of 0 as being nothing and therefore as being insignificant. In making a comparison between 0.5 and 0.25, students often ignore the zeros and conclude that, because 25 is larger than 5, 0.25 is larger than 0.5 (Resnick et al., 1989). As this example illustrates, semantic induction can be thought of as a form of intuitive mathematics, a concept inspired from intuitive physics (Chi & Slotta, 1993; diSessa, 1982, 1993; McCloskey, Caramazza, & Green, 1980). There may also be a "less-than-syntactic" component to error production, which involves failures of internal "critics," or mechanisms that signal that a rule violation has occurred (Rissland, 1985). For instance, a person may fail to develop adequate mechanisms for detecting if he or she has committed a violation. In sum, although syntactic induction is an important mechanism of error production, there is clearly a need for an account of errors that would encompass critic-related failures, through syntactic induction, to semantically based failures.

Encoding- Versus Execution-Based Errors The second limitation of repair theory and Sierra is that they only encompass errors that occur when a person reaches an impasse and tries to get unstuck by applying a repair strategy "on-line" or during the execution of problemsolving algorithms. Reaching an impasse, however, is not a necessary condition for producing rational errors. Errors not only happen during the execution of a procedure but may also occur during encoding or procedural acquisition (see also Sleeman, 1984). Thus, problem solvers who abstract an erroneous rule during learning may later execute that rule smoothly and may do so without reaching an impasse. For instance, students who give an erroneous definition of equating a limit of an infinite sequence with a bound may have constrained their definition to examples of monotonically increasing or decreasing sequences. Thus when these students are asked to provide a definition of a limit, they do so without reaching an impasse. These kinds of errors that are created during learning and do not meet impasses on new problem states are encoding errors. Indeed, Sierra agrees that repairs are induced from examples during the learning process and should thus allow for encoding-based errors that

do not meet impasses. Sierra, like repair theory, however, is an impasse-driven theory of rational errors. VanLehn (1990) explained the induction process by arguing that problem solvers incorporate all the relevant features of an example into a single, large conjunction (e.g., if only shown how to subtract on two-digit problems, the student may reason that borrowing only occurs from the adjacent and left-most digit). Later on, when the conjunction fails (e.g., on threecolumn subtraction problems where borrowing occurs from the second digit), the student reaches an impasse. Sierra does not make an explicit distinction between errors in which the conjunction fails (or is met with disconformation) and errors in which it does not. There is a need, therefore, for a clearer distinction between errors that occur during execution and those that occur during encoding. The idea that errors can also originate from the learning-acquisition phase and do not necessarily meet with impasses expands on rather than contradicts repair theory and Sierra's accounts of error production.

Domain Specificity Versus Domain Generality In addition, repair theory, Sierra, and many other accounts of rational errors tend to be domain specific (but for an exception see Davis, 1982). They thereby offer a fragmented view of error production in mathematical thinking as a whole. Repair theory, for example, focuses on subtraction errors. Other accounts of rational errors have been restricted to specific mathematical domains also, such as counting (Ginsburg, 1996), arithmetic (Ben-Zeev, 1995; J. S. Brown & Burton, 1978; J. S. Brown & VanLehn, 1980; Young & O'Shea, 1981), algebra (Matz, 1982; Payne & Squibb, 1990; Sleeman, 1984), geometry (Anderson, 1989, 1993; Dugdale, 1993), and calculus (Davis & Vinner, 1986). Conducting domain-specific research has both advantages and disadvantages. As VanLehn (1990) himself suggested, arithmetic procedures are "dry, formal, and isolated from every-day interests" and are, therefore, a "bane of teachers but a boon for psychologists" (p. 13). However, as VanLehn went on to admit, "The methodological advantages of these task domains are so great that all the major computational theories of skill acquisition address formal, common-sense-free

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task domains, even though they risk lack of generality by doing so [italics added]" (p. 13). Thus although many domain-specific accounts have provided important mechanisms of error production, such as underspecification of constraints (Matz, 1982) or faulty generalization (Sleeman, 1984) within their respective domains, they have not been integrated into more general explanations of error production.

The Need for a Taxonomy of Errors In sum, error production (a) has components that encompass more and less than syntactic induction; (b) involves encoding and executionbased errors that do not meet with an impasse; and (c) has been primarily explained by domain-specific accounts. There is thus a need for creating a more general account of errors that would incorporate the above components. The first step towards doing so is to establish a taxonomy. In the remainder of this article, I attempt to categorize errors and identify the mechanisms underlying them by integrating findings from different studies. A Taxonomy of Rational Errors A useful preliminary classification of rational errors is whether they result primarily from either critic-related or inductive failures (see Figure 2). Again, critic-related failures occur when a problem solver fails to develop internal mechanisms for detecting violations in the problem-solving process. Inductive failures occur when the person overgeneralizes or overspe-

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cializes a rule from familiar examples. Each of these components is discussed in detail in the sections that follow. Critic-Based Failures The first level of error production involves the malfunction of internal critics, or mechanisms that signal that a number-representation or rule violation has occurred. In artificial intelligence, a critic is a procedure that monitors the current problem state and signals when a constraint is violated (Rissland, 1985). The activation of a critic may therefore cause the problem solver to reach an impasse at a particular problem state. The critic can be formally represented as a production rule with a condition that lacks an action (i.e., If C then ?) and fires when it reaches an unfamiliar problem state (C). Failures may result from three kinds of situations: (a) the critic is absent, (b) the critic is weak and can thus be inhibited by a stronger prior-knowledge rule, and (c) the person engages in negation of the condition that activated the critic.

The Critic Is Absent: When a "Slip " Becomes an Erroneous Rule The absence of a critic can lead to rational errors. For example, VanLehn (1990) described a subtraction error in which students failed to decrement the digit from which they had borrowed. If such a "partially correct" error occurs only a few random times and can be corrected once it is pointed out, then it is

Unfamilar problem slate

/

1

Absent critic

Critic-related failures

Inductive failures

I \

Weak critic

Competition

Syntactic induction

Negation

I

MisPartial Spurious matching specification correlation Figure 2.

A taxonomy of rational errors.

Semantic induction

Real-life analogy

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considered a "slip," or an unintentional overlooking of a step rather than a rational error (Norman, 1981). However, as VanLehn suggested, a slip may eventually become a fullfledged erroneous rule if it is not adequately corrected. In other words, the student does not receive enough negative feedback that would lead her or him to develop an appropriate critic. Computationally, this account of error production can be modeled by deletion (VanLehn, 1990; Young & O'Shea, 1981). For example, Young and O'Shea used production rules to model correct subtraction. Then, in order to simulate students' rational errors, they omitted rules from the correct production system. For instance, they modeled the smaller-from-larger bug by omitting a rule that triggers the borrowing procedure. Once partial rules become systematic, they are extremely resilient to change (Resnick & Omanson, 1987; Rosnick & Clement, 1984). In turn, they hinder future formation of adequate critics. Thus, the absence of a rule may result in partial knowledge that is applied appropriately but still results in a rational error because there is no adequate monitoring mechanism to signal otherwise.

The Critic Is in Competition With a Prior-Knowledge Rule Even if a critic exists, it can still be overwritten or inhibited by a stronger priorknowledge rule from a different domain. The strength of a rule is primarily affected by how successfully the rule has performed in past problem-solving episodes (Anderson, 1993; Holland, Holyoak, Nisbett, & Thagard, 1986). For example, Ben-Zeev (1995) instructed college students to perform addition in a new number system called NewAbacus (see Appendix A). Participants were first given a list of NewAbacus numbers and their representation in the Base 10 system. They were explicitly taught that the digits 7 through 9 do not exist in NewAbacus. However, during the addition test, many people failed to convert these illicit numbers into their proper representations. A possible explanation is that these numbers were valid in the old and familiar Base 10 system and thus inhibited the newly formed NewAbacus number-representation critics. The fact that participants had indeed developed the adequate critics that recognized violations, such as 9 is

not a NewAbacus digit, was determined by a test of number representation where students were given a number and had to decide if it was valid in NewAbacus and to state why or why not. Competition between a current critic and a prior-knowledge rule does not result in an impasse because the critic does not reach an adequate level of activation for "firing." The third critic-related failure, on the other hand, originates from a situation where the critic actually signals a violation and is described next.

The Critic's Condition Is Negated When a person reaches an unfamiliar problemstate and the critic signals that a violation has occurred, that person may attempt to remedy the violation by simply removing it and thereby force the problem to assume its valid form. In order to remove the violation, people may negate the condition of the critic. The negation action makes the problem seem valid and therefore prevents the critic from re-signaling. More formally, when a person reaches an unfamiliar problem state (C) and the critic fires If C then ?, the person may attempt to remedy the violation by simply removing it and thereby force the problem to assume its valid form. In order to remove the violation, people negate the condition of the critic. Thus, if the critic signals If C then ?, people proceed to change the problem's form from C into ~ C . The negation action makes the problem seem valid and therefore prevents the critic from re-firing. This mechanism is different from an absent critic because the critic exists and signals a violation, but the person gets rid of the violation instead of fixing the problem. In essence, the person "fools" the critic by preventing it from re-firing. For example, in the domain of algebra, Sleeman (1984) showed that high school students tend to follow a procedure that collects all the variable terms in the equation on the left-hand side and all the numbers on the righthand side. In the process of doing so, some students get rid of "extra" multiplication signs by a process of normalization (i.e., forcing the problem to acquire what they believe is a valid or normal form). Consider the following problem-solving steps: m X X + n X X

=p

(1)

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XXXX

= p-m-

XXX=p-

n

m-n

(2) (3)

The above protocol shows that the student decides to isolate the unknown (X) by erroneously subtracting the m and the n from both sides of the equation, treating multiplication as addition. This behavior leads to too many multiplication signs on the left side of the equation (e.g., X X X X) in Equation 2. The student's critic then signals the violation too many multiplication signs, which may cause the student to negate that condition by simply removing the multiplication sign that is in excess. This action, which results in Equation 3, prevents that specific critic from re-signaling. Adults, as well as children, can engage in negating the condition of the critic. Ben-Zeev (1995) found that college students produced a deletion error on NewAbacus addition. That is, instead of changing an illicit number in an intermediate problem solving phase into its correct representation, participants simply deleted it. One may question the "rational" aspect of negation. The rationality lies in the fact that people invent a rule in order to correct a violation. Fortunately, not all errors have this form of extreme constraint satisfaction. In fact, most rational errors have been shown to arise from inductive processes that draw on the nature of worked-out examples. In particular, students have been shown to rely on a process of syntactic induction. Syntactic Induction In the process of syntactic induction, people overgeneralize or overspecialize algorithms from the surface-structural characteristics of familiar examples in a given domain. For instance, consider the following subtraction error, or "bug," called N-N-causes-borrow (VanLehn, 1986): 4

52

-12 310 The student who demonstrates N-N-causesborrow has adequately acquired the example

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that says to borrow when T < B and not to borrow when T> B. However, when the student encounters a new problem where T = B, he or she overgeneralizes the rule borrow when T < B to become the rule borrow when T ^ B. Thus, the student exhibits an innovative and systematic approach while violating mathematical principles at the same time. VanLehn (1986) discovered that only 33% of rational errors could be explained by this kind of syntactic induction.2 Additional support for syntactic induction has been provided by Ben-Zeev's (1995) empirical and computational work on rational errors in the NewAbacus number system. Specifically, after the initial instruction of the NewAbacus number representation, participants were divided into different groups. Each group received an example of a certain part of the NewAbacus addition algorithm. Then, participants were given a range of addition problems in NewAbacus, where some were familiar and some were new. The experimental investigation of the data showed that participants who received the same type of worked-out examples produced categories of similar rational errors. Similar errors were defined to be algorithmic variations of one another. For instance, one set of worked-out examples participants received illustrated how correctly to carry the digit 6. When participants in this condition reached impasses on new problems, they produced a variety of illicit carries of 6 in response (for a quick tutorial in NewAbacus, see Appendixes A and B). Using computational modeling in LISP showed that participants' rational errors were modeled best by modifying the correct procedure for the set of worked-out examples participants received. For example, illicit carries of the digit 6 were traced computationally to the example condition that illustrated how to carry the digit 6 correctly. The modifications were highly syntactic; that is, they were based on the superficial features of the examples and tended 2 In a more "liberal" analysis, one modifying the correct procedure for subtraction based on visual-numerical features of subtraction problems (e.g., top-of, left-of, bottomequal-zero, etc.), VanLehn (1986) explained 85% of second through fifth graders' subtraction errors as originating from induction. He admits, however, that this approach is more liberal because it does not show a direct link between the examples students receive in the instructional phases and the rational errors they produce.

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to create principled violations. Overall, BenZeev (1995) found that 67% of participants' rational errors were induced from examples. Syntactic induction plays an important role in error production. However, syntactic induction may be a rubric for a variety of reasoning mechanisms. Thus, I now turn to a more finegrained analysis of syntactic induction that includes the processes of partial matching, misspecification, and forming spurious correlations.

Partial Matching When a person reaches a new problem state, she or he may search for a familiar example or rule that shares some of the features of the current problem and has successfully worked in the past. Once a familiar example is found, its procedure is then implemented. This process is termed partial matching. More formally, when a person reaches a new problem state (C), and the critic signals a violation of the form If C then ?, where C itself is a composite of conditions such that C = Cx and C2 and . . . and Cn, the person searches for a familiar example or rule that contains one or more of the conditions. Once the familiar rule is found, its corresponding action (A) is fired. For instance, in NewAbacus addition, when people learn how to carry a 6 when the right digit in a NewAbacus pair is larger than 6 (see Table B1, Example 2, in Appendix B) but do not know how to proceed when the left digit in a pair is greater than 6, they know how to carry within but not between a pair of NewAbacus numbers. When they encounter a new situation where the sum of digits in a left column is greater than 6 (see Table Bl, Example 3, in Appendix B), they reach an impasse that can be formalized as follows: If there is a digit greater than 6 and the digit is a left digit then ?. During partial matching, a person who is trained on how to solve Example-2 type problems (see Table Bl in Appendix B) will look for a familiar rule and may find the following: If there is a digit greater than 6 and the digit is a right digit, then carry a 6 and leave the remainder. The first part of the familiar rule matches that of the current problem, and thus the action part of the familiar rule is executed. This notion is built on partial matching as suggested by ACT* (Anderson, 1983). If there is more than one partially matching rule that could be activated,

there is a need to select the best fitting one by a process of conflict resolution (Anderson, 1993). As I have shown, partial matching occurs when the person reaches an impasse and makes an active attempt to match the current problem with an existing one. Alternatively, if during the rule-acquisition phase the person has failed to encode the condition regarding the specific digit involved (i.e., left or right), then that person will not reach an impasse but proceed to simply execute the rule "as is." This encoding-based error is suggested to result from a process of misspecification, which is described next.

Misspecification of Constraints Rational errors may also arise as a result of misspecification of constraints during the procedural-acquisition process. That is, people may abstract rules from examples but may not adequately constrain them. These processes occur during the encoding or the procedureacquisition phase. Misspecification is best illustrated by Matz's (1982) work on the processes underlying the learning of algebraic problem solving. Matz proposed that when problem solvers encounter an unfamiliar problem state, they modify a familiar rule in order to solve the problem, a process she terms extrapolation. She offers two main examples of extrapolation techniques: linear decomposition and generalization. Linear decomposition refers to a process in which an operator is applied to each subpart of a problem independently and then the partial results from each operation are combined into a complete result. A correct use of linear decomposition can be illustrated by the distributive law of multiplication: A(B + C) = AB + AC. An incorrect use of linear decomposition can be exemplified by the following common error:

4(A + B) = VA + JB. Based on this error and similar ones, Matz (1982) suggested that people use schemata such as: U{X A Y) = \JX • D Y, which lead to erroneous solutions when applied surfacestructurally. A similar mechanism was offered by Davis, Young, and McLoughlin (1982) as cited in Rissland (1985). Linear decomposition leads to underspecified schemata where any variable or operator can fill in the missing slots. Generalization revises a new rule in order to

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fit an unfamiliar situation. For example, in factoring polynomials, the following is true:

- n)(X-m) = 0, then either (X - n) = 0 Thus X = n

or

(X - m) = 0.

or X = m.

Matz (1982) suggested that some people fail to recognize that the n and m are incidental to the problem, and can thus be replaced by any number, but that the 0 is an essential feature that cannot be replaced. Underspecification of constraints causes people to abstract the following erroneous schema:

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account of errors, Anderson's (1989) account it is not schema-based, it is presented in this section because it too emphasizes the important role of misspecification in the production of rational errors. (For more details on the role of analogical thinking in error production see Ben-Zeev, 1996.) There are other inductive mechanisms that operate during the procedural-acquisition or learning phase that can also lead to rational errors. One such inductive mechanism may be responsible for forming spurious correlations between irrelevant features in a problem and a particular rule or algorithm that solves that problem.

Spurious Correlations {X-n)(X-m)

=K

{X-n) = K or (X - m) = K X = K + n or X = K + m Anderson's (1989) account of the analogical origins of rational errors is another example of how misspecification may lead to the production of rational errors. He suggested that people correctly solve new problems by forming an analogy to familiar problems. According to the Penultimate Production System Theory of Problem Solving by Analogy (PUPS; Anderson & Thompson, 1989), examples are first encoded into declarative structures. In trying to solve a new problem, the problem solver uses the declarative structure of the familiar problem to extrapolate or map a solution. If the analogy proves successful, the declarative structure becomes proceduralized into a set of production rules. A correct analogy requires that one map one member of a category onto another (e.g., any integer can be mapped onto any other integer). Anderson (1989) suggested that there may be two origins of rational errors. The first occurs if the category in the declarative structure is misspecified (e.g., a 2 may be defined as a number instead of an integer). The second source of errors may lie in the mapping process itself (e.g., incorrectly mapping any number onto an integer). Anderson demonstrated a variety of LISP, geometry, and algebra errors that result from these kinds of analogical processes. Although, unlike Matz's (1982)

In the learning process, either verbal or written worked-out examples may contain a misleading or spurious correlation between a particular feature and a specific algorithm, which the problem solver may then abstract into an erroneous rule. A particularly compelling example comes from the vertex-intercept confusion that was presented previously. The origin of this confusion may be traced to a spurious correlation in parabolas that are symmetrical about the y axis where the y intercept and the vertex of the parabola lie on the same point. Students may have encoded this relationship into the following erroneous rule: when I am asked to find the value of the y intercept I look for the lowest or highest point in the parabola. Recently, Ben-Zeev (1998) conducted a series of experiments to examine whether people encode spurious correlations in memory, exhibit them during the learning process, and commit predictable mistakes as a result. The experiments contained a learning phase followed by a testing phase. During the learning phase, participants were instructed on how to solve problems called quantitative comparisons by using two different algorithms: multiply one side by n/n, and multiply both sides by n (see Table 1). As can be seen in Table 1, a quantitative comparison is a problem that contains two quantities: one in Column A and the other in Column B. The problem solver's task is to decide whether the quantity in Column A is smaller than, larger than, or equal to the quantity in Column B or whether the relationship between the two columns cannot be determined.

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In the experimental materials, Column A and Column B were fractions. Each algorithm (multiply one side by n/n, and multiply both sides by n) was correlated spuriously with an irrelevant feature in the denominators of these fractions. For half the participants in the study, multiply one side by n/n was correlated with a logarithm in the denominator, and multiply both sides by n was correlated with a radical in the denominator. For the other half, the featurealgorithm correlations were flipped: multiply one side by n/n was correlated with a radical in the denominator, and multiply both sides by n was correlated with a logarithm in the denominator. This manipulation ensured that the featurealgorithm correlations were arbitrary and that either algorithm was just as useful for solving problems containing either a radical or a logarithm, rendering the feature-algorithm correlations spurious. During the testing phase, participants were given a range of memory and problem-solving tasks. Results on all these tasks showed that participants preferred using the algorithm that was correlated with the irrelevant feature in a given problem for solving that problem. That is, the presence of a particular irrelevant feature cued the use of the algorithm with which it was associated during learning, leading to ineffective and at times erroneous problem solving. Another example comes from the domain of mathematical word problems. B. Ross (1984) taught college students elementary probability principles (e.g., permutation) by providing them with worked-out examples. Each example had a

particular content (e.g., involving dice). When participants were tested on the probability principles, Ross found that they associated the particular problem content with the specific probability principle with which it had appeared in the worked-out example. Thus, when the same content appeared in a problem requiring a different probability principle, participants were reminded of the original principle the content was associated with and proceeded to apply it erroneously. Relying on such a correlation heuristic, even when it is spurious, may be adaptive because it often leads to correct solutions (also see Lewis & Anderson, 1985). For instance, in school settings, a common practice of teaching arithmetic word problems is to instruct students to explicitly search for a "cue" word in a problem, and then to associate that cue with a particular solution strategy. Specifically, teachers often instruct students to associate the word left with performing subtraction on problems such as the following: "Tom has 5 apples. Jerry takes away 3. How many apples are left?" This strategy is fairly adaptive in that children quickly learn to use it on new problems. However, Schoenfeld (1988) noted that several children who are then given the word left in nonsensical word problems containing premises such as "Tom sits to the left of Jerry" proceed to subtract the given quantities in the problem, signifying that what was a well-intended strategy on the part of the teacher fostered rote learning. The idea that attending to correlational structure may be adaptive is supported by the

Table 1 Algorithms 1 and 2 and Their Correlated Features in the Quantitative Comparisons Feature Given

Algorithm 1 Multiply one column by n/n x>0 Column A Column B

x+4 log 3 Instructions

Strategy Result

2x+ 6 2 log 3

Determine whether the quantity in Column A is smaller than, larger than, or equal to the quantity in Column B or whether the relationship cannot be determined. Multiply Column A by 2/2. This gives us (2x + 8)/(2 log 3) in Column A. By comparing the 2 numerators we find that because 2x + 8 is larger than 2x + 6, then Column A is larger.

Algorithm 2 Multiply both columns by n x>0 Column A Column B x+2 2t+ 4 V5 2 ^ Determine whether the quantity in Column A is smaller than, larger than, or equal to the quantity in Column B, or whether the relationship cannot be determined. Multiply both columns by y5. This action cancels the ^5 in both columns, and leaves us with x + 2 in Column A and (2x + 4 ) / 2 o r * + 2 in Column B. Therefore, the quantities are equal.

RATIONAL ERRORS

finding that even experienced problem solvers can fall prey to spurious correlations. For example, Ben-Zeev (1998) used experienced problem solvers in her study (Yale undergraduates with high math scores on the Math Scholastic Aptitude Test). The fact that, contrary to common lore, experienced problem solvers also tend to rely on surface-structural features may be because there often is a predictive correlation between a problem's content and its deeper structure (the mathematical principles that are required for solving the problem meaningfully; see also Blessing & Ross, 1996). Overall, syntactic induction provides a rich account of error production. However, it does not tell the whole rational-error story. Next, I discuss a second level of inductive failures in which rational errors are produced by semantically based processes. Semantic Induction People appear to possess a sense of "intuitive mathematics" akin to that of intuitive physics (Chi & Slotta, 1993; diSessa, 1982, 1993; McCloskey et al., 1980). Studies in intuitive physics have shown that as a result of their experiences with the world, people develop a set of naive beliefs. For example, McCloskey et al. (1980) found that when adults were asked to draw the path of a moving object shot through a curved tube, they believed that the object would move along a curved (instead of a straight) path even in the absence of external forces. Such a conceptualization, although mistaken, is reasonable because one may draw an incorrect analogy to the Earth's circular movement around the sun (one does not "see" the forces that sustain such a movement). Although physics seems to be more applicable to events in the world, whereas mathematics appears to be more abstract, one can find similar intuitions in mathematics also. In particular, a student's mathematical intuition may draw on examples that go beyond the classroom. Such knowledge may lead the student astray because of analogical failures to real-world examples. In problem solving by analogy, the student tries to solve the current problem (the target) by retrieving a familiar problem (the source) and creating a mapping between the source and target problems (e.g., Gentner, 1983; Holyoak & Thagard, 1989; Novick & Holyoak, 1991).

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The problem arises when people form an inadequate analogy to real-life examples or concepts. In mathematics, a common error among students who start learning about exponents is that of asserting that n° = 0 rather than n° = 1. Duffin and Simpson (1993) reported that students who participated in a workshop on primes and factorization believed that 2° = 0 because, as one of the instructors had stated, "2° was no twos multiplied together, so it had to be zero." Duffin and Simpson concluded that doing nothing equals nothing is a "natural" argument that stems from real-world experience. The student may form the analogy that doing nothing:nothing.vno:O. Rational errors may also occur as a result of linguistic influences. Payne and Squibb (1990) argued that many of Matz's (1982) and Sleeman's (1984) examples of errors in algebraic manipulation problems are a consequence of such effects. For example, as was previously mentioned, Payne and Squibb argued that the precedence error [n + m X •=> (n + m)X] may result, in part, from forming a linguistic analogy such as three apples plus four gives seven apples. Davis and Vinner (1986) also suggested that language affects mathematical meaning. Davis and Vinner, remember, showed an example of students who equated the limit in calculus with a boundary that can never be reached. In terms of a linguistic explanation, Davis and Vinner argued that this confusion may originate from the fact that in everyday language the word limit does imply a boundary that cannot not be reached, such as in the phrases the outer limit of the universe and the limits of our understanding. So far the focus has been on the categories of mental mechanisms that may underlie error production. Another question altogether is the application of such a taxonomy to educational purposes. In other words, can the taxonomy be useful for understanding and correcting students' errors? Reviewing the educational literature and relating it to rational errors is beyond the scope of this article, but the following section is written in an attempt to present a few general prescriptions. Some General Prescriptions A useful prescription for remedying error production may be to provide students with "nonexamples" of a particular concept or

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algorithm (also see Shaughnessy, 1985). Workedout examples may have a crucial impact on error production. Examples may contain a high correlation between a spurious feature and a particular algorithm that the student picks up on, such as in the case of students who confused the intercept of the parabola with its vertex because they were always given examples where the two coincided. Thus, when examples contain no disconfirmatory information (e.g., parabolas in which the y intercept and vertex lie on different points), they may facilitate the production of rational errors. Nonexamples should be presented from the very beginning of skill acquisition because errors have been shown to be extremely resibent to change once they are formed (Resnick & Omanson, 1987; Rosnick & Clement, 1984). Repair theory, Sierra, and other accounts of errors emphasize the role of syntactic induction of errors. They show that students often try to solve problems by using surface- rather than deep-structural features. Thus, the second prescription is that mathematical problem solving be presented as sense-making activity (Bransford, Sherwood, & Sturdevant, 1987; Bransford & Stein, 1984; S. I. Brown & Walter, 1993; Davis & Vinner, 1986; Greeno, 1983; Resnick & Omanson, 1987; Schoenfeld, 1988, 1991) by creating classroom environments that teach for meaning rather than rote. Teaching for meaning is not a trivial task. As Schoenfeld (1991) suggested, productive teaching entails a negotiation of understanding between teacher and students. The goal is to create appropriate social environments where students are active collaborators in the learning process. One such approach to creating meaningful learning environments is S. I. Brown and Walter's (1993) what-if-not (WIN) technique. WIN approaches a problem by making it into a "situation." That is, it deletes the problem's original question and explores new questions or problems that arise from the situation. For instance, in exploring a geometrical problem, WIN would ask, "What if we were dealing with three dimensions and not with two?" Efforts aimed at challenging the givens and exploring alternative questions may help provide a richer framework for understanding the mathematics involved. It can also turn mathematical thinking into a discovery process instead of a problemsolving activity alone. Finally, remedies should be directly targeted

toward the specific mechanisms of error production. An important diagnosis to make is whether a rational error originates from the encoding or execution phase of skill acquisition. If the student reaches an impasse and then creates a solution on-line, then it means that the student has developed the appropriate critic but still needs to receive more instruction on the particular skill. If, on the other hand, the student does not reach an impasse, then the culprit may be a faulty rule that the student had previously created during the encoding or initial skillacquisition phase. The teacher then needs to help the student uncover the rule and relate it to worked-out examples the student was given during the acquisition phase. A taxonomy of errors can prove to be a valuable diagnostic as well as an instructional tool for teachers. A caveat is in order, however. Errors may be relative to a given curriculum. For example, Fuson (1992) demonstrated that students produce different arithmetic errors in Japan, Taiwan, Korea, the former Soviet Union, and the United States. Indeed, in Japan, teachers use students' common errors for instructional purposes (Stigler, Fernandez, & Yoshida, 1996). Thus, the current taxonomy of errors may be most useful as an instructional tool primarily for U.S. teachers. More work needs to be dedicated toward uncovering the commonalties and differences among errors across different cultures. Such an investigation could yield more insights on domain-specific versus domain-general components of procedural learning. In sum, rational errors do not have to be a hindrance to the mathematical learning process. They can also serve as constructive and adaptive tools for promoting understanding (see also Smith, diSessa, & Roschelle, 1993). That is, in the process of "debugging," or searching for the origin of one's errors, the student may reach a better understanding of his or her own mathematical reasoning. By committing errors and understanding their origins, students may achieve a stronger conceptual basis for reasoning correctly than if they had never committed the errors in the first place. Discussion: What Can Be Learned From a Taxonomy of Rational Errors? As this article has shown, a person often meets the challenge of solving a new mathematical problem state by creating rule-based but

RATIONAL ERRORS

erroneous algorithms that lead to rational errors. These erroneous algorithms are the problem solver's attempt to create a reasonable solution in a relatively short period of time with minimal computational cost. In essence, the person's efforts are directed toward interpreting and adapting to the mathematical environment in much the same way that he or she would adapt to a social or physical environment. That is, the person's cognition can be characterized by a "bounded rationality" (Simon, 1957), aiming to achieve what seems to be a viable rather than an optimal solution. This article supports the idea that there are a few mechanisms underlying the seemingly diverse collection of rule-based errors. The new taxonomy posits that rational errors can be categorized as resulting from critic-related failures, through syntactic misinduction, to semantic misinduction. Such a taxonomy adds to existing accounts that suggest that rational errors are primarily produced by syntactic induction or "symbol pushing" alone. The less-than-syntactic component, or criticrelated failures, emphasizes the role of internal critics in error production. The more-thansyntactic component contains errors that result from semantic induction or intuitive mathematics. Similarly to intuitive physics, mathematical knowledge may also draw on a person's experience with real-world examples. Regarding the conceptualization of syntactic induction itself, even when a person engages in symbol manipulation without regard to underlying principles, often his or her performance makes probabilistic sense. Worked-out examples most often contain a high co-occurrence between a particular feature and a specific algorithm without providing enough, if any, disconfirming instances where the same feature co-occurs with a different operator. Therefore, from a probabilistic perspective, it is adaptive for a person to create a spurious featurealgorithm correlation. Spurious correlations and other erroneous rules that are formed during the encoding or skill-acquisition phase and do not meet with impasses are overlooked by accounts such as repair theory and Sierra. The idea is that such erroneous rules are induced primarily from examples and are then added to the problem solver's collection of all available rules for the skill. When a relevant problem state is activated, the spurious rules may be executed smoothly.

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On the other hand, in accordance with repair theory and Sierra, I have discussed mechanisms that are created on-line, or during the execution stage of problem solving. For instance, negation occurs as a result of an impasse or when a critic fires a violation or detects an unfamiliar problem-state. The present taxonomy, therefore, expands on rather than contradicts repair theory and Sierra's impasse-and-repair account of error production. The discussion of the mechanisms underlying rational errors has provided answers to several questions but leaves others open. For example, to what extent do rational errors in mathematics share commonalties with rational errors in other specific domains such as physics, statistics, and logic, as well as with broader domains such as language acquisition, social cognition, and reasoning? For instance, one may wish to explore the commonalties between mathematical and linguistic errors. Specifically, do both types of errors share similar mechanisms and developmental patterns? Karmiloff-Smith (1986) argued that there exist developmental shifts in language acquisition ranging from procedural through meta-procedural (i.e., marked by an awareness of one's own rules) to conceptual phases. In the shift from the procedural to the meta-procedural phase, for instance, young children tend to make grammatical errors on verbs they had previously acquired correctly, such as erroneously adding the past form -ed to roots of familiar verbs before developing a more permanent and correct knowledge base. The parallels to an account of mathematical errors may be that (a) the child applies linguistic rules systematically rather than randomly and therefore produces rational errors, and (b) the child overgeneralizes from correct rules of grammar and thus produces inductive-based errors (also see Pinker, 1994). In contrast to linguistic errors, however, rational errors in mathematics do not appear to follow a clear developmental pattern where children versus adults engage in more syntactic performance that, in turn, provides a stepping stone for subsequent development of more conceptual representations. In fact, adults' performance on arithmetic in a new number system (Ben-Zeev, 1995) mirrors young children's performance on addition and subtraction problems. Both populations rely heavily on syntactic

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induction and engage in constraint satisfaction such as negating the condition of the critic. In order to investigate the generality of error-production mechanisms and to further explore error production in mathematical reasoning, there is need for new methodologies. Currently, empirical work is sorely lacking in research on the production of rule-based errors. For the most part, the investigator "waits" for a person to commit an error. Only then, in a post hoc fashion, does the investigator proceed either to computationally model the error, or to provide mechanisms underlying its origin, or both. In contrast, it may be useful to "force" people into making errors. In other words, in order to understand truly the origin of errors, researchers need to predict a priori the kinds of errors that people will create in a given domain before people actually commit them. Finally, there are important educational questions to pursue. Specifically, how can knowledge about mathematical error production be used toward teaching students the correct acquisition of mathematical concepts and procedures? The prescriptions section has offered several strategies for achieving this aim, such as using one's own errors as a vehicle for acquiring a better understanding of correct problem solving. However, unless we fully understand the theoretical underpinnings of rational errors in mathematical thinking first, people will continue correctly, meticulously, and creatively to follow their incorrect rules. References Anderson, J. R. (1983). The architecture of cognition. Cambridge, MA: Harvard University Press. Anderson, J. R. (1989). The origin of errors in problem solving. In D. Klahr & K. Kotovsky (Eds.), Complex information processing: The impact of Herbert A. Simon (pp. 343-371). Mahwah, NJ: Erlbaum. Anderson, J. R. (1993). Rules of the mind. Hillsdale, NJ: Erlbaum. Anderson, J. R., & Thompson, R. (1989). Use of analogy in a production system architecture. In S. Vosniadou & A. Ortony (Eds.), Similarity and analogical reasoning (pp. 267-297). Cambridge, England: Cambridge University Press. Ashlock, R. B. (1976). Error patterns in computation. Columbus, OH: Bell & Howell. Beck, A. T. (1967). Depression: Causes and treatment. Philadelphia: University of Pennsylvania Press. Beck, A. T. (1985). Theoretical perspectives on clinical anxiety. In A. H. Tuma & J. D. Maser

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Appendix A The NewAbacus System I have constructed a new number system based on the Base 10 abacus system, which is a written version of number representation on the abacus. This new number system is called NewAbacus (see Figure Al), and is described in more detail in Ben-Zeev (1995). In the NewAbacus system each Base 10 digit is represented as two digits. The left digit can either be 6 or 0, whereas the right digit can range from 0 through 5. The Base 10 digit is represented by the sum of left and right digits. For example, 8 in Base 10 is equivalent to 62 in NewAbacus (6 + 2 = 8). Although 64 and 65 in NewAbacus sum up to be 10 and 11 in Base 10 respectively, they are "illegal" in new abacus because 64 and 65 violate the basic number-representation rule that requires each Base 10 digit to be represented by two digits in NewAbacus. The correct representations for 10 and 11 in NewAbacus are 0100 and 0101, respectively.

0 = 00

10 = 0100

20 = 0200

1 -01

11=0101

30 = 0300

2 = 02

12 = 0102

40 = 0400

3=03

13=0103

50 = 0500

4 = 04

14 = 0104

60 = 6000

5 = 05

15 = 0105

70 = 6100

6 = 60

16 = 0160

80 = 6200

7 = 61

17=0161

90 = 6300

8 = 62

18 = 0162

100 ==010000

9 = 63

19 = 0163

Figure Al. Base 10 numbers (to the left of the equals sign) and their representation in new abacus (to the right of the equals sign). From "The Nature and Origin of Rational Errors in Arithmetic Thinking: Induction From Examples and Prior Knowledge," by T. Ben-Zeev, 1995, Cognitive Science, 19. p. 350. Copyright 1995 by Cognitive Science Society Incorporated. Reprinted with permission.

383

RATIONAL ERRORS

Appendix B Addition in NewAbacus Table Bl Examples That Illustrate Parts of the NewAbacus Addition Algorithm Example noAule

Example

Action —

6002 + 0202 6204

1. No carry

2. Carry into 6

+ 05

"5

Invalid number Cany a 6, leave a remainder

63

3. Carry from 6

4. Carry into and from 6

0462 + 0161 0323 0333 0305

Sum colums, cany a 10 Add digits Form valid number

16 63 + 0405 «

Invalid number, so carry a 6 and leave remainder Sum columns, carry a 10 Add digits Form valid number

0522 0523 0504

Note. From "The Nature and Origin of Rational Errors in Arithmetic Thinking: Induction From Examples and Prior Knowledge," by T. Ben-Zeev, 1995, Cognitive Science, 19, p. 351. Copyright 1995 by Cognitive Science Society Incorporated. Adapted with permission.

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The NewAbacus addition algorithm is divided into four main parts. They are (a) no carry, (b) carry into the 6 digit, (c) carry from the 6 digit, and (d) carry into and from the 6 digit. Each part is illustrated by an example (see Table Bl). In the no carry example, NewAbacus addition is identical to addition in Base 10. Addition is done column by column without any carries. In the carry into the 6 digit example, the addition results in an intermediate solution where the right digit in a pair is equal to or greater than 6. This step, therefore, results in a violation. The correct algorithm is to carry the 6 to the left and leave the remainder. For example, when the right column in the intermediate solution is 9, one carries a 6 and leaves a remainder of 3. Note that carrying a 6 only occurs within a pair of NewAbacus digits. In the carry from the 6 digit example two 6s are added in one column to produce a sum of 12. In this case, one carries a 1 to the next pair, and leaves a remainder of 2. Thus, a carry of 1 only occurs between a pair of new abacus digits. However, one should still contend with Ihe 2 in the left digit because it violates the left-digit rule (i.e., a left digit can only be 6 or 0). The correct algorithm is to sum the 2 with the right digit to form a valid NewAbacus pair. Finally, the carry into and from the 6 digit example is simply a combination of the last two cases.

Received September 23, 1997 Revision received March 11, 1998 Accepted April 5, 1998

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