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Algebra Students and Rational Expressions 1 Running Head: ALGEBRA STUDENTS AND RATIONAL EXPRESSIONS Algebra Students’ Simplification of Rational Expressions Hanna Figueras, Lorraine Males and Samuel Otten Michigan State University Algebra Students and Rational Expressions 2 Algebra Students’ Simplification of Rational Expressions Even algebra students who are adept at simplifying a variety of polynomial expressions often face a significant challenge when confronted with rational expressions—the quotient of two polynomials. There are many possible reasons for this. Perhaps students do not have a solid comprehension of rational numbers and so are unable to handle something that looks like a "fraction with x's in it." Perhaps they are unsure of the order of operations since there may be operations both "above" and "below" the operation of division. Perhaps they have inadequate conceptual understanding of inverse operations and so inappropriately cancel terms. Or perhaps it is something else entirely. Nevertheless, this is important because errors in this domain, according to our experience, are frequent, and algebra students will likely see rational expressions again later in their algebraic career (and certainly if they go on to calculus). The purpose of this research was to examine how algebra students simplify rational expressions, thus contributing to the existing research on algebra learning. Although rational expressions have not been the direct focus of much research, the domain promises to provide a rich supply of data for several reasons. First, experience leads one to expect student errors to be plentiful. Second, the complexity of rational expressions suggests that the strategies students use to simplify them are likely to be diverse. Third, there are strong visual cues in many rational expressions as objects are placed in close vertical or diagonal proximity to one another. Fourth, rational expressions may be the first entity in algebra for which students are not given or are unable to generate a concretization to correspond with the expression. These Algebra Students and Rational Expressions 3 points imply that early algebra students, even those progressing successfully through the material, may face their first serious roadblock in rational expressions. Much of the past work regarding algebra learning has focused on student errors during algebraic manipulation (e.g., Matz, 1980; Demby, 1997). A recurring theme in these studies has been reliance by students on the visual cues present in algebraic structures. A visual cue is anything in the structure of a mathematical object that brings to mind a piece of mathematical knowledge or a memory of a mathematical experience. An illustration of this would be a student who sees 3( x 2) and recalls the distributive property based on the visible structure of a number situated adjacent to parentheses. Reliance on visual cues is not necessarily undesirable since skilled algebra students can use visual cues efficiently, appropriately, and successfully. However, visual cues may lead many algebra students to commit errors. For example, Demby (1997) wrote about the 3 and -3 appearing in the expression 2 x 3 3x as being visually attractive, often leading students to cancel inappropriately. Kirshner (1989) found that some students depend on visual cues rather than propositional reasoning, and that students who lack a propositional foundation for their syntactic knowledge often retrieve inappropriate rules. Rational expressions present strong visual cues, with like terms (or identical terms) often situated vertically or diagonally from one another, as in 3x 6 . 3x This expression may prompt students to recall the cancellation property ab b . If students a lack a firm propositional foundation (Kirshner, 1989), then they may inappropriately cancel Algebra Students and Rational Expressions 4 the 3x in the numerator and denominator, as was found by Matz (1980) who referred to it as the cancellation error. It is important to note that a student committing such a cancellation error may indeed possess the corresponding propositional knowledge but may simply have failed to activate this knowledge in the face of such strong visual cues. Do certain presentations of rational expressions make it more likely for a student to access propositional knowledge during simplification? With this question in mind, we turn to literature that regards mathematical entities as processes and objects. According to Sfard (1991) there are two different ways in which to conceive mathematical notions: operationally (as processes) and structurally (as objects). It is especially difficult for early algebra students, transitioning from arithmetic to algebra, to conceive of polynomials as objects rather than processes. For example, students may see the expression 3( x 5) 1 as a process or a sequence of instructions, such as: add five to a given number, multiply the result by three, and then add one to this result (Sfard & Linchevski, 1994). Similarly, Gray and Tall (1992b) wrote about the proceptual divide. A procept is “a combined mental object consisting of both process and concept in which the same symbolization is used to denote both the process and the object which is produced by the process” (p. 4). For example, 3 2 signifies the counting process and the product of this process, namely, 5 . Thus, seeing 3 2 conjures up both the process of addition and the concept of sum. According to Gray and Tall (1992), the proceptual divide occurs between students who can “successfully see the algebraic notation as flexible procept and those who see it only as process and fall into routines using instrumental procedures to carry out computations” (p. 6). Algebra Students and Rational Expressions 5 This notion of process and object can be found in many other works as well, though often with different terminology. Freudenthal (1983) wrote about the difference between procedural and static interpretations, the former pronouncing a b as “a plus b” and the latter viewing a b as the sum of a and b. According to Kieran (1992), Matz and Davis have also studied this idea, calling it the process-product dilemma. Regardless of the language used, this conceptual framework has implications for rational expressions. Students who commit cancellation errors may be viewing the rational expression as a process to be carried out, with cancellation being an expected part of this process. Students who interpret a rational expression (at least to some extent) as a quotient of two objects may be less likely to commit cancellation errors because such a cancellation involves an inappropriate parsing of the objects. In addition, there may be a relationship between the object conception of rational expressions and the propositional knowledge needed to avoid cancellation errors. A presentation of rational expressions that emphasizes the object interpretation, then, would be expected to negatively correlate with cancellation errors during simplification. Research Questions This study examined algebra students’ simplification of rational expressions with an emphasis on the relationship between cancellation errors and an object interpretation of rational expressions. We focused on the following research questions: 1. What errors do algebra students commit when simplifying rational expressions? 2. What strategies do algebra students use to simplify rational expressions? Algebra Students and Rational Expressions 6 3. What is the effect on the frequency and types of errors committed when rational expressions are presented in a way that affords an object interpretation? Method Study Design To investigate students’ errors and strategies in simplifying rational expressions, we administered a survey to groups of students in grades 9, 11 and 12. The survey comprised two parts. Part 1 consisted of 7 items designed to determine base levels of cancellation errors committed. Part 2 consisted of 4 items designed to present rational expressions with an emphasis on an object interpretation. (See the appendix for a complete list of the survey items.) Participants The study included 80 students. Thirty-six were 10th-grade students in a geometry class in a rural public high school in northern Michigan. Forty-four were 11th- or 12th-grade students in a pre-calculus class in a rural/suburban public high school in central Michigan. All students had completed at least one year of a course designated as algebra. All had studied rational expressions prior to the time that the survey was given, thus having had an opportunity to learn how to simplify rational expressions. Instruments The survey consisted of 11 problems divided into two parts. The parts were known to the researchers, but the orders of the problems were randomized so that neither part preceded the other thus preventing the students from learning from previous survey items. Part 1 problems (there were 7) presented rational expressions in the variable x to be simplified. The Algebra Students and Rational Expressions 7 coefficients and constants in the expressions were positive integers. Part 2 problems (there were 4) presented rational expressions, each of which was designed in a different way to evoke interpretation as mathematical objects. The first used parentheses, the second identified the numerator and denominator with visual symbols, and the third and fourth presented situations in which the numerator and denominator could be readily associated with something either conceptually or visually. Each problem in Part 2 corresponded to one of the problems in Part 1. The first two problems in Part 2 were varied so that they corresponded with distinct problems from Part 1. Procedure With the permission of the principals and teachers, the survey was given cooperatively by teachers and researchers to the students in their mathematics classrooms. The northern surveys were administered in December of 2007 and the central surveys in January of 2008. The students were asked to fill out the survey in pen and show their work. The use of pens would prevent students from changing answers to earlier questions as they progressed through the survey. The students were given approximately 20 minutes to complete the survey. Analysis The raw data consisted of responses to 80 surveys with 11 items in each survey. One item was discarded from analysis (the reason for this will be discussed later in this paper). In addition, five surveys were discarded due to a large number of uncodable items (see below for definition of uncodable). Thus, the final analysis included a total of 750 items from 75 surveys. Categories of errors were developed based on the student work, frequencies of these Algebra Students and Rational Expressions 8 errors were tabulated, and categorical analysis was conducted on responses to the linked items from Part 1 and Part 2. Categorization of Errors Initial analysis entailed coding student responses for types of errors. We identified the following primary error categories: cancellation (C), partial division (P), like-term error 1 (T1), like-term error 2 (T2), linearization (L), defractionalization (D) and equationization (E). These seven categories accounted for 83% of the errors and are described in detail below, with examples presented in Table 1. It should be noted that 24% of the items containing errors were double-coded because participants, for example, made a visual cancellation error in the first step of their simplification and a like-term error in the second step. In addition to these seven main error categories we used three additional codes. The other error (OE) category consisted of any algebraic error that was not included in the primary categories; for example, a factorization mistake such as 5 x 15 5( x 2) or a misconception such as x x was coded as OE. An item on which the participant progressed x through the simplification without committing an error, whether or not it was a complete simplification, was coded as no error (NE). Finally, blank or undecipherable response and responses in which the participant merely rewrote the problem were categorized as uncodable (UN). Cancellation (C). The characteristic feature of a type C error was the visible presence of an algebraic literate—a constant term, a variable, a coefficient, and so forth—in both the numerator and the denominator that was cancelled by the participant. This cancellation was usually done with a written slash, but work was also classified as C if the participant simply Algebra Students and Rational Expressions 9 removed the visible terms in the subsequent step. Furthermore, a participant canceling terms after rewriting the rational expression to make visible the terms to be cancelled was also coded as C. Partial Division (P). Type P errors were characterized by division taking place, but only between some of the terms in the rational expression. For example, in 3x 9 3x 3 , 9 3 was divided by 3 to yield 3, but the denominator was not divided into the 3x term. (Note that a similar example appears in the table as C because in that instance the student rewrote the expression to make visible the 3 with which to cancel the denominator). Another example was 4 x 20 x 20 4x x , in which we saw that was reduced to , but this was not division by 12 4 x 3 4x 12 3 the entire denominator. Like-Term Error 1 (T1). Participants committing the first like-term error performed an operation other than division (usually subtraction) between terms in the numerator and terms in the denominator. An instance of T1 was 20 8c 20 6c , in which we saw the 2c difference of the like terms 8c and 2c . Like-Term Error 2 (T2). Participants committing the second like-term error performed a mistaken operation with terms in the numerator or performed a mistaken operation with terms in the denominator. This included erroneous addition of like terms or inappropriate addition of unlike terms. For example, 7x x 8x 2 was classified as T2. 7 x 3x 10 x 2 Algebra Students and Rational Expressions 10 Linearization (L). This error, noted in Matz (1980) as well as other places, was characterized by the breaking up of a rational expression with a compound denominator into two separate rational expressions, as in 4 4 4 . 8 4x 8 4x Defractionalization (D). This error is characterized by the transformation of a fraction with unitary numerator to a nonfraction, such as 1 1 x or 3 . x 3 Equationization (E). Transforming a rational expression into a rational equation was classified as E. For example, a participant operating within this category would convert the expression to an equation and then conclude their work with x 3 . Note that merely writing the equals sign between steps in the simplification process was not classified as E. The inclusion of the equals sign had to alter the content of the item to warrant an E classification, such as setting the rational expression equal to 0. Not having anticipated an error where students subtract individual terms in the denominator from individual terms in the numerator, we had created the item expression was often incorrectly simplified as follows: 10 4x . This 2x 10 4x 10 2x . According to our 2x coding scheme, this error could be categorized as T1 or P because, based on our analysis of 4 other survey items, students could be either subtracting ( 4 2 2 ) or dividing ( 2 ). 2 Therefore, this item was discarded from analysis. Results We found a larger quantity and a wider variety of errors in the survey responses than we had expected. In fact, 79% of all 750 items were not simplified correctly and 74% of the Algebra Students and Rational Expressions 11 attempted items contained at least one error. Table 2 displays the frequencies of error types across all items. As we had anticipated, the most common error was the cancellation error (24% of all codes). In fact, 51 of the 75 participants committed the cancellation error. Partial division and the unexpected like-term error 1 were also relatively common. Forty participants made the partial division error, and interestingly, 38 of these 40 students also committed the cancellation error. Also, many students made the same error on several items: 21 students (28%) committed errors of the same type on at least 7 of the 10 items, and 33 (44%) committed errors of the same type on at least 6 items. Because of Part 2’s focus on object interpretation of expressions, we had anticipated fewer errors in Part 2 than in Part 1. Part 2, however, contained a large proportion of uncodable items (36%), whereas Part 1 only contained 8% uncodable (see Table 3). This made it difficult to draw any conclusions on how the construction of Part 2 problems might have afforded an object interpretation and how this correlated with simplification success. Table 4 presents a chi-square frequency distribution of the responses to the linked items. Based on McNemar’s test for marginal homogeneity, the only significant conclusion that can be drawn from the data is that students who committed an error in Part 1 were likely to either make an error or not attempt the corresponding item in Part 2 (p<0.00001). A large portion of the students simply left the word problems blank. Discussion We examined high-school algebra students’ simplification of rational expressions and categorized their errors. Students were less successful in their simplifications than we Algebra Students and Rational Expressions 12 anticipated, even though all had received prior instruction on simplification of rational expressions. Nearly four-fifths of the items were not simplified correctly. There was evidence that students were using consistent strategies as they worked to simplify the rational expressions, but these strategies were most often incorrect. As was predicted, we did see an apparent reliance on visual cues, for instance, when students would strike out identical terms during a cancellation error or would rewrite a term so that they could see a factor and cancel it across the division bar. Additionally, students who consistently committed the cancellation error, partial division, or the linearization error may have been using a strategy that was based on a flawed conception of division. The existence of a commonality behind some of these error types was evidenced by the fact that 95% of the participants who committed partial division also committed the cancellation error. However, our study design did not allow us to draw conclusions regarding the thought processes underlying the apparent strategies. One thing that we did not anticipate but that showed up very powerfully in the data were strategies based on a fixation with like terms. This was seen most clearly in the large number of like-term 1 errors wherein students performed operations only between like terms, whether or not the terms were together in the numerator or denominator, and often students’ work was completely void of any division operation. In other cases (i.e., partial division), students performed division but only between like terms rather than between the entire numerator and denominator. This suggests that partial division may be at the intersection of a misconception of division and an over-attention to like terms. Algebra Students and Rational Expressions 13 Our attempt to draw a comparison between the Part 1 and Part 2 items failed due to an apparent lack of engagement in Part 2. This may have been a deficiency of the paper-andpencil format we implemented. A design that included a clinical interview may have been more likely to elicit responses on problems like those in Part 2. Moreover, such a design would allow for information to be gathered regarding the thought processes underlying the errors and strategies that we observed. It would also be beneficial for future studies to include the like term errors in their theoretical framing, since our data pointed significantly to this phenomenon. For teachers, the primary implication of this study is that high school algebra students have difficulty simplifying rational expressions. This is especially significant because simplification is a first step in working with rational expressions, and errors in this domain will make it nearly impossible for students to be successful in subsequent work, for instance, solving rational equations or rational inequalities. Teachers need to be aware of the common errors that students make when simplifying rational expressions, especially the cancellation error since this may be frequent. This study also suggests that teachers be aware of their emphasis on like terms during instruction. Though it is clearly necessary to cultivate in algebra students an awareness of like terms, it is also equally necessary for them to understand when and when not to use like terms as the basis for simplification strategies. Algebra Students and Rational Expressions 14 References Demby, A. (1997). Algebraic procedures used by 13-to-15-year-olds. Educational Studies in Mathematics, 33(1), 45-70. Freudenthal, H. (1983). Didactical Phenomenology of Mathematical Structures. Dordrecht: D. Reidel Publishing Company. Gray & Tall (1992). Success and failure in mathematics: Procept and procedure. In Workshop on Mathematics Education and Computers, (pp. 209-21), Taipei: Taipei National University. Retrieved from http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1992lsecondary-procept.pdf. Kieran, C. (1992). The learning and teaching of school algebra. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning, (pp. 390-419). New York: MacMillan Publishing Company. Kirshner, D. (1989). The visual syntax of algebra. Journal for Research in Mathematics Education, 20(3), 174-187. Linchevski, L. & Livneh, D. (1999). Structure Sense: The relationship between algebraic and numerical contexts. Educational Studies in Mathematics, 40(2), 173-196. Matz, M. (1980). Towards a computational theory of algebraic competence. Journal of Mathematical Behavior, 3, 93-166. Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1-36. Algebra Students and Rational Expressions 15 Sfard, A. & Linchevski, L. (1994). The Gains and the Pitfalls of Reification: The Case of Algebra. Educational Studies in Mathematics, 26 (2/3), 191-228. Algebra Students and Rational Expressions 16 Table 1 Examples of student errors Error Student Examples Categories Cancellation Partial Division Like-Term 1 Like-Term 2 Linearization & Other Error Algebra Students and Rational Expressions 17 Table 2 Error Frequencies Error Frequency (n = 557) Cancellation error (C) 0.24 Partial division (P) 0.16 Like-term 1 (T1) 0.19 Like-term 2 (T2) 0.08 Linearization (L) 0.09 Defractionalization (D) 0.05 Equationization (E) 0.03 Other error (OE) 0.17 Algebra Students and Rational Expressions 18 Table 3 Comparison of the Two Parts Part I Part II n = 450 items n = 300 items 371 error codes 186 error codes No error (NE) 0.24 0.17 Uncodable (UN) 0.08 0.36 0.76 0.83 0.74 0.74 Erroneous or uncodable items (over total) Erroneous items (over attempted) Algebra Students and Rational Expressions 19 Table 4 Chi-square distribution of the 225 linked items Part 1 No error Errors Uncodable No error 32 8 2 Errors 9 96 5 Uncodable 7 56 10 Part 2