1 Lorraine M. Males Analytic Summary of Kirshner, D. (1989). The visual syntax of algebra. Journal for Research in Mathematics Education, 20(3), 174-87.
Research Problem According to Kirshner (1989) the ability to understand the syntactic structure of an algebraic expression is fundamental to the understanding of elementary algebra. In order to factor and evaluate expressions students must be able to parse equations correctly by recognizing the dominant features of an expression. For example, when a student is asked to factor the expression 2 x2 10 , they must recognize subtraction as the dominant (least precedent) operation of the expression and that the x is squared before the 2 is multiplied. To be able to successfully perform transformations such as factoring and evaluating students must be able to successfully parse equations. However, students’ syntactic knowledge may be depending on visual cues, rather than propositional rules (rules of operations). Since students may be depending on these visual cues in their syntactic decision making, rather than propositional reasoning, they may be disadvantaged when retrieving the appropriate templates to solve problems. Hence, students will use rules for simplifying rational expressions involving multiplication to simplifying rational expressions involving addition. Another problem is that although research has noted the importance of syntactic skill, curricular and pedagogical practice has been inattentive to it. Textbooks do not devote much time to discussing algebraic syntax and when it is discussed, it is often inadequate, sometimes leaving out important ideas such as exponentiation and radicals.
2 Purpose The purpose of this study was to determine how syntactic knowledge of algebraic notation is initially encoded and instantiated. Theoretical Perspective The question about the nature and origin of students’ syntactic knowledge led Kirshner to consider three hypotheses each based on a different perspective. The work done by Davis (1984) introduced the idea of a visually moderated sequence. For example, a visual cue elicits a procedure whose execution elicits a new visual cue and so on. This idea would mean that the retrieval of a procedure is connected to the visual cues one would receive and therefore ones’ syntactic knowledge may be encoded visually. Students’ syntactic knowledge may be instantiated by the “look” of the expression, such as the raised exponent and the space between the terms rather than by the propositional content (the operations). This perspective became the basis of Kirshner’s hypothesis 2. The work of Anderson (1983), on the other hand, provided a different perspective. Anderson identifies a process of specialization, which he called proceduralization. This process occurs “through the instantiation of the general parameters of the declarative knowledge structure using the particular characteristics of the task domain” (Kirshner, 1989, p.275). The propositional rules are instantiated and then the student creates a procedural representation, which is represented visually. This model became the basis of Kirshner’s hypothesis 1.
3 Kirshner chose to look at a third perspective, which became the basis for his hypothesis 3. The third perspective was that the visual hierarchy does not enter into syntactic process and therefore there is no change in the representation. As a way of obtaining evidence of visual syntax a nonce (or artificial) language was formed that displayed the propositional content of algebra, but distorted the spacing and used letters to represent operations. For example, 3MxE2 represented 3x2 . The ability to parse an expression in unspaced nonce would indicate the presence of propositional knowledge, whereas the inability to parse this expression would indicate that students were relying on the visual cues of regular notation. Research Questions and/or Focus Kirshner seeks to determine which of the following hypotheses regarding the initial encoding and instantiation of syntactic knowledge of algebraic notation is true: Hypothesis 1- Syntactic knowledge initially acquired in propositional form becomes represented in a visual modality. Hypothesis 2 - Syntactic knowledge initially acquired in a visual modality comes to be represented in propositional form. Hypothesis 3 - There is no change in the modal representation of syntactic knowledge. (Kirshner, 1989, p.275) Method Written expression evaluation tasks were given to 404 students in 15 classes. Students in grades 9 and 11 were from two middle and lower-middle class public schools in Vancouver, Canada and freshman calculus students were from the University of
4 British Columbia. Twenty-three students were removed from the sample because they did not meet the arithmetic criterion. Kirshner refers to the group of subtests as the instrument. The instrument originally contained three 10-item subtests. The first subtest included only nonce arithmetic items, such as 3E2. The second subtest included five simple nonce items, such as 3MxA4 (unspaced) or 3 M x A 4 (spaced), and five complex nonce items, such as 1A3MxE2 (unspaced) or 1 A 3 M x E 2 (spaced) in an arbitrary order. The third subtest included regular notation algebra items that corresponded in syntactic structure to the nonce items. For example, since the seventh item on the nonce test was 1A3MxE2, the seventh item on the regular notation test was 3 2x 2 . The multiple-choice nonce tasks all included just one instance of the variable x and for each task the students were asked to evaluate the expression for x = 2. Each task included as choices all the possible parsings of the expression and a blank for other possible responses. For example, 1A3MxE2 had response choices of 13, 16, 37, 49, 64, ____. Written in regular notation this would be 1 3x2 .The responses were given to accommodate all possible parsings such as: 1 3(2) 2 13 , (1 3) 22 16 ,
1 (3 2) 2 37 , (1 3 2) 2 49 , and ((1 3) 2) 2 64 .
A fourth subtest needed to be added in order to determine if students were visualizing the nonce items in regular notation. Students were asked to think about one of the nonce tasks he or she had answered correctly and respond yes, no, or maybe to a question that asked them if they had visualized this nonce expression in regular form. For example, one question was “For the problem 1A3MxE2, did you imagine or visualize or picture in your mind 1 3x2 ? Students were also asked if they used the rules of order of
5 operations, and if so, which rules. These were called self-reports and were used to judge whether successful completion on the nonce tasks was due to visualizing this notation in regular-notation. In order to prevent students from changing the nonce notation into regular notation students were not allowed to use scratch paper. They were instructed to write down and circle only their final answer and to use pen (to eliminate erasures). Two versions of the instruments where created by varying the nonce notation (spaced or unspaced) used. The subtests were randomly distributed within gender groupings in each class.
Findings & Evidence The mean score of the regular notation algebra subtest was 92.3%, the simple nonce notation was 89.2%, the complex nonce notation was 60.5%. For each of the subtests except the complex nonce subtest, there was no significant difference between genders or those using the unspaced and spaced nonce notation. However, on the complex nonce subtest, more women had difficulty with the unspaced nonce notation. The analysis showed that students who were unable to apply propositional rules most often processed the symbols from left to right. For example, most students (85) obtained the correct response of 13 for 1A3MxE2. The response that received the next highest number of responses was 64, which comes from parsing the expression from left to right [(1A3)Mx]E2 or [(1 3)( x)]2 . This parsing from left to right was the most frequently chosen distracter. More than half of the incorrect responses were due to this distracter.
6 Analysis of the self-reports showed that success on nonce notation tasks was not based upon visualizing these tasks in regular notation. Of the students who either responded no or yes to the question of whether they were using a visualization strategy, about half of them said that they were not. Analysis of the regular notation tasks indicated that almost all students could evaluate 1 3x2 for x 2 . Students were either able to transfer their skills to the nonce notation or they were not. For the students whose regular notation skills transferred, it was evident that it was more difficult for students to transfer this ability to the unspaced nonce notation than to the spaced nonce. It was concluded that “for some students the surface features of ordinary notation provide a necessary cue to successful syntactic decision” (Kirshner, 1989,p.282). The self-reports also indicated that success on the nonce tasks is related to the students possessing appropriate propositional rules of algebraic syntax. Students whose regular notation skills did not transfer, Kirshner believed supports the hypothesis that syntactic knowledge is acquired initially in a visual modality. This seems to be strong evidence against hypothesis 3 because there was a change. Students were able to evaluate an expression in regular notation, but were not able to evaluate a similar expression in nonce notation. However, it is not clear to me that because their skills did not transfer that they initially acquired the syntactic knowledge visually. Kirshner even states that it is possible that these students did initially acquire the syntactic knowledge in proportional form earlier, but may not be able to retrieve it anymore.
7 Kirshner argues that the theoretical, curricular, and historical evidence is more compelling than the empirical evidence. It is not clear that this study was able to provide evidence of this type. However, the argument that Kirshner puts forth does seem valid and has implications on the teaching of algebra. He states that hypothesis 2, which involves the syntactic knowledge initially acquired visually coming to be represented propositionally, requires simpler cognitive implementation. He believes that the experiences required for hypotheses 1 and 3 are not currently being attended to in schools. Finally, Kirshner states that “the presence of systematic visual features in algebraic notation that correlate with propositional categories alerts us to unconscious processes in algebraic syntax.” Conclusions/Implications What implications does Kirshner’s study have? First, according to this study, syntactic knowledge must also be linked to cognitive representations that are not visually present. Students must be able to correctly parse an expression, which is visually present, in order to do simple derivations, but must also access semantic templates that are not present for visual reference, but are in long-term memory. According to Kirshner, The student who can successfully analyze an expression presented on a page, but who has no propositional basis for syntactic knowledge, may be severely disadvantaged in the retrieval of appropriate templates. Many common errors, like
ax a (Matz as cited in Kirshner, 1989), would seem to be instances of the b x b
retrieval of correct but inappropriate rules, in this case
ax a (p. 285). bx b
8 Even though almost all students in this study were able to parse an algebraic expression, only some developed propositional skills. For these students without propositional skills, visual cues provided the basis for their syntactic knowledge. Kirshner believes that curriculum that does not emphasize the propositional syntax of algebra is providing students with incomplete information and is hindering their retrieval of appropriate transformational rules. If students do not learn to use rules that enable them to retrieve information that is not visually present, they will often have difficulty retrieving these rules.
Overall Assessment Reading Kirshner’s article was interesting, but difficult. The complex nature of the problem makes it difficult for me to know whether Kirshner’s method was in fact proving anything. The development and use of the nonce notation added an element of complexity that makes it hard for me to use as a model. I believe that this research is significant because it is attempting to look at how students are thinking and what might be triggering their actions in performing algebraic transformations. Kirshner’s article helps to inform our group research project. It is possible that students are having difficulty retrieving the appropriate templates for simplifying rational expressions. This article has enabled me to see that this retrieval problem may be linked to the reliance on visual cues. References Kirshner, D. (1989). The visual syntax of algebra. Journal for Research in Mathematics Education, 20(3), 174-87.