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Jl. of Comput Computers in Mathemat ematics and Science Teaching (2007) 26(3), 181-196

Computer as a Medium for Overcoming Misconceptions in Solving Inequalities SERGEI ABRAMOVICH State University of New York at Potsdam USA [email protected] AMOS EHRLICH Tel Aviv University Israel [email protected] Inequalities are considered among the most useful tools of investigation in pure and applied mathematics; yet their didactical aspects have not received much attention in mathematics education research until recently. An important aspect of teaching mathematical problem solving at the secondary level deals with the notion of equivalence of algebraic transformations used in replacing inequalities by equations. This article is motivated by computer-enhanced activities designed for and carried out with prospective teachers of mathematics. It shows that the appropriate use of computer graphing software has the potential to avoid errors and overcome misconceptions associated with the notion of equivalence in solving inequalities. The article demonstrates how mathematical visualization made possible by powerful yet user-friendly graphing technology provides learners with a conceptual insight into the sources of errors typical for the secondary mathematics classroom.

The current reform of school mathematics curricula and pedagogy is driven by the fundamental belief that all students should learn important

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mathematical concepts and procedures with understanding. Technology has great potential to play the major role in the realization of this belief. The National Council of Teachers of Mathematics (2000) has expressed its vision of the reform through a number of principles and standards. It has been suggested that whereas mathematical proficiency required for the 21st century workplace develops through conceptual understanding, the appropriate use of technology enables students’ success in dealing with complex problems that otherwise were inaccessible to them. Indeed, technology “influences the mathematics that is taught and enhances students’ learning” (National Council of Teachers of Mathematics, 2000, p. 11). Such a focus of the reform requires high levels of problem-solving skills on the part of the students and their teachers alike. One of the most difficult topics in secondary school mathematics deals with the notion of equivalence in solving inequalities. Nonetheless, according to the standards, all students in grades 9-12 should “understand the meaning of equivalent forms of…inequalities and…solve them with fluency…using technology in all cases” (National Council of Teachers of Mathematics, 2000, p. 296). These ambitious expectations for the students raise the level of professional standards for secondary mathematics teachers from their current position. Therefore, of particular importance is the development of pedagogical strategies that make it possible for the teachers “to establish a discourse that is focused on exploring mathematical ideas” (National Council of Teachers of Mathematics, 1991, p. 52) in identifying common errors and misconceptions associated with the notion of equivalence in solving inequalities. In what follows, a number of strategies that can help the teachers to establish and enhance such discourse by using technology will be presented. This article, in part, reflects on the first author’s experience in teaching technology-rich mathematics education courses to preservice secondary school mathematics teachers. The courses address a number of recommendations by the Conference Board of the Mathematical Sciences (2001) for teacher preparation, including the creation of a new capstone sequence that emphasizes the role of computer applications as tools for exploring complex algebraic ideas and discovering connections among them. In particular, within such a course sequence, prospective teachers of secondary mathematics can learn how these ideas “underlie rules for operations on expressions, equations, and inequalities” (p. 40). Through the appropriate use of technology the teachers can discover how these rules change as one proceeds from solving an equation to solving an inequality. It should be noted that while inequalities are among the most important tools of pure and applied mathematics, their didactical aspects have received

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little attention until recently. At the 28th annual meeting of the International Group for the Psychology of Mathematics Education inequalities and their significance in the precollege mathematics curriculum was at the focus of the Research Forum titled Algebraic Equations and Inequalities: Issues for Research and Teaching (Boero & Bazzini, 2004). In particular, it has been recognized that an inquiry into the appropriate use of technology in the conceptualization of students’ difficulties (or, simply, errors) in the context of inequalities is a legitimate direction of mathematics education research (Tall, 2004). Furthermore, it has been rightly suggested that these errors may be due to students’ inadequate experiences with solving equations. As Balacheff (1990) stated earlier, “errors are not mere failures but symptoms of specific pupils’ conceptions” (p. 262). ERRORS AND THEIR ORIGINS The studies of errors in problem solving originally concerned the diagnosis of arithmetical errors (Ashlock, 1986). Different theories were formulated to explain some stable errors and enable learners to recognize and overcome misconceptions (Brown & Van Lehn, 1980; Sleeman, 1984; Sackur-Grisvard & Leonard, 1985; Batanero, Godino, Vallecillos, Green, & Holmes, 1994). The rapid expansion of computer technology has given birth to different intelligent tutoring systems (Sleeman & Brown, 1982; Thompson, 1989; Yerushalmy, 1991; Schwartz, 1993), which provided an effective alternative to traditional methods in diagnosing errors in arithmetic and algebra. In the context of precalculus, Wenger and Brooks (1984) used a computer as a diagnostic scaffolding tool in a verbal guiding of the equivalence in transforming inequalities. Radatz (1979) argued that errors often arise from what he called the negative transfer (alternatively, Einstellung effect)—a phenomenon when experience with similar problems results in “habitual rigidity of thinking” (p. 167). Matz (1982) reiterated this argument by showing that many errors in high school algebra occur “as a result of a systematic adaptation of previously acquired knowledge using a small number of extrapolation techniques” (p. 27). A naïve view of similarity between equations and inequalities—two relations that on the surface level differ in one symbol only—gives birth to an extrapolation technique responsible for the occurrence of that type of errors. More recently, Tsamir and Bassini’s (2004) research showed that many precollege students believe that “solving inequalities and equations are the same process.” In particular, these authors raised a question of

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whether computer-based instruction can improve students’ proficiency in solving inequalities and called for research to address it. Such an improvement begins with the preparation of teachers. The present article suggests that mathematical visualization based on the appropriate use of computer graphing software has the potential to provide prospective teachers with effective strategies and easy-to-implement techniques of reducing an inequality solving to the solving of the corresponding equation(s) enabling one to avoid errors associated with the Einstellung effect. EQUATIONS VERSUS INEQUALITIES The Technology Principle formulated by the National Council of Teachers of Mathematics (2000) is based on the assumption that technology influences what and how mathematics can be taught to students and therefore, effective teaching should capitalize on “selecting or creating mathematical tasks that take advantage of what technology can do efficiently and well— graphing, visualizing, and computing” (p. 26). Goldenberg (1991), an early advocate for the use of graphing utilities in teaching school mathematics, argued for the importance of students’ learning to interpret graphs as a way of overcoming misconceptions and eliminating confusions. However, as mentioned by Eisenberg and Dreyfus (1991), many students prefer procedural approaches to problem solving and are reluctant to accept benefits of visualization as a tool of mathematical understanding. In the area of inequality solving, technology makes it possible to combine procedural skills with computer-enhanced visualization thus helping learners of mathematics overcome specific difficulties associated with the notion of equivalence. In order to solve an inequality, one has to coordinate different ideas, both algebraic and geometric, associated with the properties of functions involved. Recent research on the teaching and learning of inequalities (Boero & Basini, 2004; Tsamir, Tirosh, & Tiano, 2004) indicated that students who are skilled in arithmetic computations, simplifying algebraic expressions, and even equation solving, may have enormous difficulties and experience a great deal of confusion when solving inequalities. Trying to extend their facility with equations to inequalities, they lack knowledge of the rules of equivalence when transforming both sides of an inequality. For instance, given an inequality, students tend to disregard the issue of equivalence when multiplying or dividing its both sides by the same variable expression (Tsamir & Bazzini, 2004). In contrast to solving equations—a task which students often are able to master even without conceptual understanding of


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the transformations involved (Steinberg, Sleeman & Ktorza, 1990)—the violation of the transformations for inequalities may lead to a solution set that looks stunningly different when compared with the correct one. That kind of phenomena can be observed and conceptualized through mathematical visualization enhanced by computer graphing. ANOTHER SOURCE OF ERRORS In the early 1990s, the authors had the opportunity to examine the assumption that secondary school students’ misconceptions in solving inequalities can be ascribed to an inadequate knowledge of their teachers in this area (Abramovich & Ehrlich, 1993). A sample consisted of 300 mathematics teachers, newcomers to Israel, representing a wide range of ages, instructional experiences, and nationalities. These teachers were presented with a written test consisting of the 7-12 level mathematics problems, including nonlinear inequalities in one variable. It turned out that on the average, irrespective of background, examinees experienced the same difficulties and produced similar types of errors as those typically demonstrated by secondary school students. In particular, these errors resulted from multiplying and dividing both sides of inequalities by a common variable expression. This finding led the authors to believe that the reason for the persistence of students’ misconceptions in the area of solving inequalities stems, to a large extent, from inadequate training mathematics teachers received as part of their preservice/inservice education. This is consistent with Ashlock’s (1986) comprehensive analysis of children error patterns in arithmetic that suggested these errors are based on either flawed or insufficiently learned mathematical concepts. By appropriately integrating technology into mathematics education courses for secondary teachers one can improve teacher preparation in the context of solving inequalities. COMPUTER GRAPHING AS A MEANS OF VISUALIZATION The utilization of a computer as a medium for visualization in the context of graphing has been known for more than two decades. The practice of using computer-programming languages, such as BASIC, as means of plotting graphs gradually shifted towards the use of computer applications that do not require extensive learning of syntax and semantics of a programming language. Nowadays, the approach to “teaching with” the computer—a ref-

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erence to which one can find as early as in the work by Demb (1973)—is considered across different educational documents as the most effective way of engaging learners into exploring mathematical content (Shaw, 1997; International Society for Technology in Education, 2000). In this article, the authors suggest using the Graphing Calculator 3.5 (referred to as the GC)—a computer application produced by Pacific Tech (Avitzur et al., 2004)—both as a support system in solving inequalities and as a cognitive tool in illuminating misconceptions. The ease of the tool’s functioning and its pedagogical effectiveness in the context of solving equations with parameters was reported elsewhere (Abramovich, 2005; Abramovich & Norton, 2006). The uniqueness of the GC is that it can graph a relation from any two-variable equation or inequality. Reflecting on a mathematics education course content taught by the first author with the GC, prospective teachers of secondary mathematics often relate this software to the Equity Principle (National Council of Teachers of Mathematics, 2000) by acknowledging “any student can easily use [it because] … the list of commands to master is straightforward and simple” and affirming that the tool “ensures more student involvement [as it]…could be very helpful for those struggling with basic concepts.” In a more specific context, a preteacher noted that the tool “allows students see equations and inequalities in another way…without using algebra or at least with using only some algebra.” An important implication of these comments is that the use of the GC may be extended beyond mathematics teacher education programs. As another preteacher (looking forward to her own teaching) put it: “It is probably better than anything else we have been using. Advantages outweigh limitations. Very useful tool in the classroom—will be used next year.” This unique computational capability of the GC to graph relations can be utilized to graph the solution of any inequality in one variable in the form of a horizontal bar (or a set of bars) “penetrated” by the x-axis (see Figures 1-4). For example, in order to graph the solution of the inequality f( f x)>g(x), one can construct a two-variable inequality (1) and graph it in the (x, y)-plane of the GC for a relatively small value of ε. Such a solution graph would have the form of a horizontal bar (or bars). Indeed, the radical in inequality (1) squeezes any vertically infinite strip (or a set of strips) defined in the (x, y)-plane as the set of points {(x,y)| f( f x)>g(x), -∞x2. It should be noted that although the last inequality is not difficult to solve either without technology or by using any available graphing utility, the use of the GC plays an instructive role here as the authors follow a pedagogical principle of using the didactical transparency of simple examples in communicating complex mathematical ideas. However, the graphing of a two-variable inequality that defines the solution graph of the inequality x>x2 would not be possible without the use of the GC—a GC tool capable of graphing relations. Of course, one could write a program in, say, BASIC programming language that enables such complex graphing; yet, in the context of mathematics teacher education, that would be a futile shift from the current practice of “teaching with” the computer to the past practice of “teaching about” the computer. The same pedagogical principle will underlie the authors’ choice of examples and their computer-based geometric representations in the rest of the article. To conclude this section note that the psychological underpinning of the proposed graphical (geometric) method of reducing inequality to equation(s) can be drawn on the work by DeSoto, London, and Handel (1965). These authors argued that a subject who is told that A is greater (smaller) than B forms an image of A above (below) B because the relations of greater and smaller have a fixed link to the vertical axis in human cognitive space. This implies that learners are good at thinking of objects as spatially ordered. Therefore the translation of an inequality into its geometric representation based on the above-below relations between the inequality’s left- and right-hand sides has the potential to provide cognitive support to learners in solving inequalities. Furthermore, the use of visual imagery makes it possible to reduce inequality solving to finding solutions of the corresponding equation(s) from which the endpoints of intervals that comprise the solution set of the inequality result. In such a way, a learner can apply a previously

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studied concept (equation solving) to a new domain (inequality solving) using a method free of errors that might result from the Einstellung effect.

Figure 1. Solving the inequality x>x2 by constructing its solution graph CONCEPTUAL SOURCES OF ERRORS IN SOLVING INEQUALITIES An interesting educational task is to establish conceptual sources of errors arising from the application of nonequivalent transformations to algebraic inequalities; in other words, “not only to eliminate such errors but to identify what their origin might be” (Balacheff, 1990, p. 262). The GC can be used as a medium for identifying and conceptualizing different outcomes of transforming inequalities including extension, reduction, and preservation of the solution set. The software enables one to see the relationship between both sides of an inequality in the “above-below” format. Furthermore, this relationship is reflected through a simultaneous construction of the solution graph on the x-axis of the (x, y)-plane. This makes it possible not only to diagnose possible errors but better still, to provide explanatory feedback in conceptualizing the very sources of these errors.


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Quite often, inequalities that one encounters at the secondary level include fractional expressions with variable denominators. Having experience with multiplying fractions in arithmetic by a common denominator amplified by, generally speaking, an imperfect practice of eliminating fractions in the case of equations (e.g., consider a nonequivalent transfer from the equation f( f x)/g )/ (x)=h(x) to that of f( )/g f x)=g(x)h(x) when there exists x0 such that f x0)=g(x0)=0), one frequently (and quite naturally) attempts to multiply both f( sides of an inequality by a variable denominator being unaware of the hidden complexity of such an operation. That is, by extending this operation from the domain of fractions and fractional equations to that of inequalities, one can hit upon the Einstellung effect. While in the case of equations such a transformation (as it was previously mentioned) might lead to an extraneous solution, which could be easily identified, in the case of inequalities one encounters multiple results: preservation, extension, reduction, as well as both extension and reduction of the solution set. In much the same way, canceling out a variable expression in both sides of an inequality is, generally speaking, a nonequivalent operation. Once again, this practice might stem from one’s imperfect experience with equations (e.g., consider a nonequivalent transfer from the equation f x)g(x)=f f( )=f( )=f f(x)h(x) to that of g(x)=h(x) when there exists x0 such that f( f x0)=0). While in the case of equations such an operation can lead to the loss of a solution only, in the case of inequalities multiple outcomes can be observed. Illustration 1: Effects of multiplying both sides of an inequality by a variable expression In order to explain the first type of the above-mentioned phenomena in general terms, consider the inequality h(x)< f( f x)/g )/ (x) )/g (2) A novice practice, stemming from one’s (correct) experience in eliminating fractions in arithmetic, often results in the inequality h(x) g(x)