Taxonomy for assessing conceptual understanding in algebra using multiple representations.
The concept of unknown and operations with unknowns are central to teaching and learning middle school algebra. Students often reveal a certain degree of proficiency manipulating algebraic symbols, and when encouraged, can verbalize and explain the steps they perform, thereby demonstrating awareness of procedures according to fixed rules. It is well documented however that correct and seemingly fluent demonstration of a procedure does not necessarily indicate conceptual understanding (Herscovics, 1996; Herscovics & Linchevski, 1994; Kieran, 1992; Kieran & Chalouh, 1993; Langrall & Swafford, 1997).
To objectively analyze and assess students' thinking and behaviours, and make informed decisions about learning algebraic concepts beyond procedures, educators need reliable indicators to refer to, and simple, yet structured language to communicate. This paper describes a segment of a large scale longitudinal study on representations in algebra and suggests the taxonomy for assessing conceptual understanding of linear relationship with one unknown.
Three key ideas serve as the foundation of this research study: a) the role of multiple representations in probing understanding of mathematics learning, with a particular focus on the interpretation, connection and translation among representations of the structurally same relationship, b) the idea of adaptation to abstraction and the development of conceptual understanding as a process of growth of the degree of abstraction, and c) the idea of reducing the level of abstraction as a mental process of coping with abstraction level of a given concept or task. These ideas guided the researcher's observations, analysis, and formulation of the taxonomy.
This section briefly outlines major theoretical positions and research that pertained to and served as a background for this research study.
Algebraic reasoning and conceptual understanding in algebra
Algebraic reasoning has been described as a process of generalizing a pattern and modeling problems with various representations (Driscoll, 1999; Herbert & Brown, 1997). Langrall and Swafford (1997) defined algebraic reasoning as the "ability to operate on an unknown quantity as if the quantity is known" (p. 2).
Understanding in general and in mathematics in particular is a logical power manifested by abstract thought. Conceptual understanding in algebra is demonstrated by the ability to recognize functional relationships between known and unknown, independent and dependent variables, and to discern between and interpret different representations of the algebraic concepts. It is exemplified by competency in reading, writing, and manipulating both number symbols and algebraic symbols used in formulas, expressions, equations, and inequalities. Fluency in the language of algebra demonstrated by confident use of its vocabulary and meanings as well as flexible operation upon its grammar rules (i.e. mathematical properties and conventions) are also indicative of conceptual understanding in algebra.
Symbol systems and representations are essential to mathematics as a discipline since mathematics is "inherently representational in its intentions and methods" (Kaput, 1989, p. 169). Goldin and Shteingold (2001) suggested to distinguishing internal and external systems of representation (p. 2).
Internal representations are usually associated with mental images people create in their minds. Pape and Tchoshanov (2001) described mathematics representation as an internal abstraction of mathematical ideas or cognitive schemata, that according to Hiebert and Carpenter (1992) the learner constructs to establish internal mental network or representational system. Thus, internal representation and abstraction are closely related mental constructs.
According to Goldin and Shteingold (2001), an external representation "is typically a sign or a configuration or signs, characters, or objects" and that external representation can symbolize "something other than itselt" (p. 3). Most of the external representations in mathematics (e.g., signs of operations, symbols, or composition of signs and symbols used to represent certain relationships) are conventional; they are objectively determined, defined and accepted (p. 4).
Many researchers (e.g., Boulton-Lewis & Tait, 1993; Diezmann, 1999; Diezmann & English, 2001; Outhred & Saradelich, 1997; Verschaffel, 1994; Swafford & Langrail, 2000) agree that in mathematics education the term representation refers to the construction, abstraction and demonstration of mathematical knowledge, as well as illustration of problem solving situations. Mathematical relationships, principles, and ideas can be expressed in multiple structurally equivalent representations including visual representations (i.e., diagrams), verbal representations (written and spoken language) and symbolic representations (numbers, letters). The abilities to recognize, create, interpret, make connections and translate among representations are powerful communication tools for mathematical thinking. Each representational system contributes to effective communication of mathematical ideas by offering a certain type of language to express mathematical ideas in a precise and coherent way, thus providing multiple sources and avenues to develop conceptual understanding of mathematics. Dreyfus and Eisenberg (1996) suggested that a fluent and flexible use of multiple representations of "structurally the same" (p. 268) mathematical concept is likely to be associated with deep conceptual understanding.
The research focused on students' generated representation and subsequent impact of these representations on learning mathematical concepts suggests that when students generate representations of a concept or while solving problems (as a means of mathematical communication) they natural tend to reduce the level of abstraction (given by the concept/problem) to a level that is compatible with their cognitive structure (Hazzan, 1999; Hazzan & Zaskis, 2005; Pape & Tochanov, 2001). Similarly, Wilensky (1991) asserted that it is expected that students would make the unfamiliar more familiar, and the abstract more concrete. He argued that students try to 'concretize' the concepts they learn to "come to the concept as close as possible" (p. 196). The process of 'concretizing' can be associated with the construction of an internal representation and can involve the process of reducing the level of abstraction. Therefore, it seems logical to view representations and reducing abstraction as closely related ideas.
Abstraction as a mental action separates a property or a characteristic of an object from the object to which it belongs or is linked to and forms a cognitive image or a concept (an abstraction) of the object. Thus, abstraction can be understood as a mental process that promotes the basis of thoughts that allow one to reason.
The concept of abstraction in the field of mathematics education research has been examined from different perspectives (Ferrari, 2003; Frorer et al., 1997; Gray & Tall, 2007; Heibert & Lefevre, 1986; Ohlsson & Lehtinen, 1997; Skemp, 1986; Tall, 1991). Hazzan and Zazkis (2005) assert that certain types of concepts are more abstract than others, and that the ability to abstract is an important skill for a meaningful learning of mathematics. Mathematics students are continuously involved in the process of abstraction because they are engaged in transformation of their perceptions into mental images by means of different representations. Essential to this research are the notion of the degree of abstraction (Cifarelli, 1988; Heibert & Lefevre, 1986; Skemp, 1986; Wilensky, 1991), the notion of adaptation to abstraction (Piaget, 1970; Von Glaserfeld, 1991), and the notion of reducing level of abstraction (Hazzan, 1999; Hazzan & Zaskis, 2005).
Cifarelli (1988) suggests the levels of reflective abstraction that include recognition, representation, structural abstraction, and structural awareness. At the highest level (i.e., structural awareness) the student is able to grasp the structure of the problem and to represent solution methods without resorting to lower levels of abstraction.
If the degree of abstraction is a factor of conceptual understanding, then the idea of adaptation to abstraction becomes critical, and the process of building mathematics conceptual understanding can be viewed as a transition between the levels of abstraction from lower to higher. Thus the growth in conceptual understanding is manifested by the increased ability to "cope with" (Hazzan, 1991; Hazzan & Zaskis, 2005) a higher degree of abstraction.
The following example uses the idea of the levels of abstraction as a metaphor to describe the process of developing conceptual understanding in algebra. Assume that operating on 'number words' which represent certain quantities of real objects is a first level of abstraction (linguistic abstraction). Then, operating with 'number symbols' can be thought as the second level of abstraction, and operating on letters that stand for 'number symbols' can be viewed as the third level of abstraction (algebraic abstraction). Thus, one can assert that abstraction in mathematics is an activity of integrating pieces of information (facts) of previously constructed mathematics knowledge and reorganizing them into a new mathematics structure. The Table 1 shows the transition from concrete (number system, pictorial aids) to abstract (algebraic symbols).
It is important to recognize that a line segment image of a number and/or unknown, as any external representation, provides limited information, and "stresses some aspects and hide others" (Dreyfus & Eisenberg, 1996, p. 268). Yet, this representation might be sufficient to supplement and enhance the process of building the concept of operations with unknowns.
To describe learners' behaviors in terms of coping with levels of abstraction, Hazzan (1999) introduced and Hazzan and Zazkis (2005) elaborated on a theoretical framework of reducing level of abstraction. The framework addresses the situations in which students are unable to deal with the concepts at the level they are presented with and therefore, the students reduce the level of abstraction to make these concepts mentally accessible (p.102). It seems plausible to assume that every algebra student goes through the process of familiarization with and adaptation to different levels of abstraction at a different rate. Wilensky (1991) suggested that the higher the rate of adaptation to abstraction the less the need for reducing the level of abstraction. In this sense, the process of adaptation to abstraction involves certain behavior manifested in coping with level of abstraction. In other words, when stu dents are unable to manipulate with the level of abstraction (words, numbers, symbols) presented in a given problem, they consciously or unconsciously reduce the level of abstraction of the concepts involved to make these concepts within the reach of their actual mental stage of development (Vygotsky, 1985, pp.84-86).
The above overview of the ideas and assumptions about representations, abstraction and conceptual understanding provided the basis for developing the study that offered another perspective on the process of assessing algebra students' conceptual understanding of linear relationship with one unknown.
A multi-year mixed method research study was launched to explore the levels of middle school students' understanding of linear relationship with one unknown. The research inquiries have been addressed through analysis of the survey and observation of students' thinking process while solving problems and explaining their solutions during the interviews.
The survey, designed by the researcher (Panasuk, 2006), consisted of several interrelated parts, four of which are described in this paper. Part I contained 12 items with five optional Likert scale response choices (always, often, sometimes, rarely, never), which were clustered around students' preferred mode of representation (verbal, pictorial, symbolic). Part II statements had three choices and asked the students to select the response that most closely reflects their current learning practices and most preferable/less preferable mode of thinking (words, diagram, numbers/symbols) when solving linear equations with one unknown.
[FIGURE 1 OMITTED]
Part III illustrated "structurally the same" (Dreyfus & Eisenberg, 1996, p. 268) linear relationship with one unknown posed in three different representations: as a word problem, as a diagram where the unknown quantity was presented as a line segment, and as an algebraic equation (see Figure 1).
The students were not asked to solve the problems, but rather to observe and explain in writing if they recognize structurally the same relationship (i.e., the sum of 10 and unknown number is 28) presented in three different modes.
Part IV had three sets of problems: Set W (words), Set D (diagrams) and Set S (symbols). Each set consisted of three problems that involved linear relationships with one unknown to be solved using one-two-step addition/subtraction and multiplication/division. The students were asked to solve each problem. Set W had three word problems that could be modeled by means of linear equations with one unknown. Set D posed three linear relationships presented in visual form via diagrams similar to problem (b) on the Figure 1. Set S contained three linear equations with one unknown represented in symbols, similar to the problem (c) on the Figure 1. The problems in each set had their counterparts in other sets presented in different modalities. For each part of the survey and for each Problem Set (W, D, and S) coding systems were created.
During the period of four consecutive years four tiers of data were collected from 11 schools in four suburban and two large underperforming urban districts with diverse populations of students. The schools were not randomly selected but were approached by the researcher with a request for participation in the study. All the participating schools used the same mathematics curriculum, which claimed facilitation of reasoning skills and use of multiple representations. The schools administered the survey to all 7th and 8th algebra students (Ntotal = 753). Each of the four tiers of surveys was analyzed and compared to describe students' ability to recognize structurally the same linear relationship presented in different representational
modalities and the ability to solve problems posed in words, diagrams and symbols. The analysis prompted the researcher to organize all surveys in three distinct groups to form three major categories, which in turn induced generation of a hypothesis about the indicator of student conceptual understanding of linear relationship with one unknown. The categorization of the surveys guided the selection of the students (N=18) for the interviews, six students from each group. The interviews were centered around the students' abilities to 'cope with' the level of abstraction presented by the linear equations with one unknown and to recognized the same relationship presented in three different representational modes. The researcher was also observing how the students i) extracted information from situation and were able to represent the information in different modes, ii) manipulated representations, and iv) interpreted and tested the solutions of the linear equations with one unknown.
Based on the analysis of the students' abilities to solve linear equations with one unknown (Part IV, Problem Sets) and their abilities to recognize structurally the same relationship presented in different modalities (Part III), the taxonomy for assessing conceptual understanding in algebra using multiple representations was formulated.
Phase 0. All 1 the students who formed this category reported in Part I and Part II of the survey that they preferred memorizing the rules and remembering the steps, thinking 'in numbers', and needed to use 'trial and error' when solving problems. They did not recognize that three different problems (Part lid represented structurally the same relationship. Seventy-seven percent of the students in this category found the unknown number correctly for the problems presented in words (Part IV, Problem Set W), only about half (56%) of these students found the unknown length of the segment for the problems presented in a diagram (Problem Set D), and 86% found correctly the unknown number in the algebraic equation (Problem Set S). Overwhelming number of this category students used trial and error method to find the solutions.
During the interview, these students were asked to create an algebraic equation of the relationship stated in words ('the sum of two numbers is 23, one of the numbers is 9, find the other number). They either showed subtraction in a column format, or wrote the numerical equation 23 9 = 14. None of the students in this category produced a symbolic statement beyond the level of numbers that would describe the relationship where a letter stands for unknown number (e.g., x + 9 = 23). These students apparently had difficulty operating with algebraic sentences (equations) and preferred numerical instantiations (Kieran, 1992, p. 392).
Interestingly enough, the students in this category either were having difficulty or were unable to solve the equations that represented the same relationship in a pictorial mode. As a result, their lacking ability and possibly not favorable attitude toward pictorial representation created a barrier for their meaningful learning (Kieran, 1992) and consequently development of conceptual understanding. Further probing revealed great confusion with diagrams when they were asked to interpret a given diagram or to create their own. For example, when asked to draw a diagram that would represent subtraction of 9 from 23, some drew 23 objects (squares or circles), then drew 9 more of the same objects, and then said they would take away 9.
Summarizing the above, the persistent need of trying the numbers (trial and error method) when coping with problems involving linear equations with one unknown showed that the students made an effort to reduce the level of algebraic abstraction (letter symbol) to numerical abstraction (number symbol). The dependency on the numbers was an indicator that these algebra students still needed instantiations to operate at the comfort level that numbers provided to them. One may hypothesize that for these students the process of adaptation to the numerical abstraction has achieved some endurance, and that numbers have become their internal representation of the material objects. However, an algebraic symbol (letter) that stands for an unknown in a linear relationship was yet an external representation that had not been internalized and integrated into the students' prior mental structure, and thus it was rejected in favor of the lower level of representation (number symbol) which had been a tool of their mathematical communication mastered to date (Javier, 1987; Lesh, et. al., 1987). It is possible that they developed computational skills to the degree of being able to reproduce and/or mimic the procedure. However, their algebraic reasoning (as defined by Driscoll, 1999; Herbert & Brown, 1997; Langrall & Swafford, 1997) had not been advanced to the level of abstraction given by the symbolic representation of the concept. Therefore, it is highly unlikely that these students achieved conceptual understanding of the linear equation with one unknown.
Phase 1. The students who formed this category reported that they didn't need to try the numbers when solving equations, thus didn't need to reduce the level of abstraction given by the problem. Eighty six percent of the students in this category indicated that they could 'think in symbols' and rather preferred symbols to diagrams and word problems. The students also reported that they preferred using the steps when solving equations (82%), and were able to memorize rules (93%). Sixty eight percent reported that didn't need or want to use diagrams when solving problems, however when presented with a diagram that displayed a linear relationship with one unknown were able to find the length of the unknown segment. Eighty nine percent of the students in this category solved all nine problems included in the survey Part IV. Nevertheless, these students either did not recognize (indicated 'no' in the survey) that the three different representations in Part HI posed structurally the same relationship, or left this part blank. Some students in this group attempted to describe their thinking, but did not produce clear written explanation in their surveys, and when interviewed were not able to verbally explain the connections between the representations. When pro vided with the diagram (see Fig. 2), they could find the length of the unknown segment. However when asked to write an algebraic statement that would describe the relationship presented by the diagram, they produced the same type of numerical statement (e.g., 23 - 9 = 14) as the first category of students.
[FIGURE 2 OMITTED]
These students knew how to perform and explain the steps while solving linear equations with one unknown, which was consistent with their survey responses in Part I and Part II. They found the correct numerical value for unknowns, and even substituted the values to the equations to verify the solutions. Yet, their behavior could have been described as a well rehearsed acting upon fixed rules (e.g., isolate the unknown, undo or use inverse operation). Probing questions revealed that these students' actions were rather mechanical than rooted in logic. These observations support the theory that many middle school algebra students (particularly those who are in transition from pre-algebra to algebra) learn procedural skills before developing conceptual understanding (Dubinsky, 1991; Dubinsky & McDonald, 1991; Kieran, 1992; Sfard, 1991, 1992).
In summary, the students who were able to manipulate symbols, verbalized and followed the correct steps when solving linear equations with one unknown, and showed certain degree of proficiency without reducing the level of abstraction, were likely to be in a process of developing conceptual understanding. It is also likely that they were in the path of blending the procedural and conceptual knowledge (Tall, 2008). They showed relatively fluent reproductive skills (i.e., process skills) which are prerequisite for the development of conceptual understanding. However, these students were missing one essential capability. They were not able to make connections between different representations (words, diagrams, symbols) that posed structurally the same linear relationship. It is important to stress that the researcher does not claim that the Phase 0 and Phase 1 are sequential and represent a hierarchy. The students who were at the Phase 1 might not necessarily have gone through Phase 0.
Phase 2. All the students in the third category recognized (answered "yes"; survey Part III), and explained that the word problem, the diagram and the equation (see Fig. 1) represented structurally the same linear relationship. They solved all nine problems in Part IV correctly and revealed understanding of meaning of solutions and the properties of the linear relationship. When probed with questions, they showed the ability to explain the full meaning of the concept of unknown, its relationship to the operations, and demonstrated the ability to discern, infer and interpret different representations of the linear relationship with one unknown. It was evident that these students were able to manipulate different representations and demonstrated flexible thinking of the properties of the linear equations with one unknown (e.g., reflection, symmetry, equivalence). When asked to describe the diagrams similar to those shown on the Figure 2 using algebraic symbols, they produced algebraic equations (e.g., x + 9 = 23). When asked to draw diagrams that would represent equations (e.g., m + 13 = 38, 2n = 26, 3x + 2 + 4 = 27), they generated correct representations. According to Cifarelli (1988), these students operated at the higher level of reflective abstraction, i.e., structural awareness.
Given the above, it is plausible to assert that one of the most significant indicators of conceptual understanding of linear relationship with one unknown is the ability to recognize structurally the same relationship presented in different representational modalities, provide an explicit verbal explanation, and flexibly transit from one representation to another.
Emerged from this research, the taxonomy (see Table 2) of the conceptual understanding in algebra was developed. This taxonomy is in line with the theories that advocate multiple representations and distinguishing levels of abstraction (e.g., Boulton-Lewis & Tait, 1993; Cifarelli, 1988; Diezmann, & English, 2001; Hazzan, 1999, Hazzan &Zaskis, 2005; Hiebert & Lefevre, 1986; Skemp, 1986).
Acknowledging the distinction between the levels of abstraction in the process of development of conceptual understanding, seems useful practical tool for mathematics educators. The indicators of a conceptual understanding would be demonstrated by the level of "structural awareness" (Cifarelli, 1998), ability to operate upon "object conception" (Sfard, 1991; Dubinsky, 1999) without necessity to reduce the level of abstraction presented by the problem (Hazzan, 1999; Hazzan & Zaskis, 2005).
The taxonomy provides teachers, educators, curriculum specialists, and other interested parties with some organizational structure for them to be able to make a relatively reliable judgment as to whether the students are developing or have developed conceptual understanding of linear equations with one unknown. The very fact that a student recognizes that the concept/relationship can be presented in different modes might serve as an indicator that the student is advancing from procedural skills to structural or conceptual skills. Thus, the taxonomy helps to ascertain whether students are building conceptual understanding instead of efficiently repeating the process. It also might provide the teachers with insight into the level of each student's thinking process and the student's way of operating with
abstractions inherent in algebra. Such information is essential to planning instruction for naturally diverse population of students with wide range of abilities, learning preferences and attitudes.
Of course, the cognitive processes of abstraction and developing conceptual understanding are much more complex than taxonomy. Any schematization has its natural limitations. As Raymond Nickerson (1986) noted, "Taxonomies are, at best, convenient ways of organizing ideas and should never be taken very seriously. The world seldom is quite as simply divisible into neat compartments as our penchant for partitioning it conceptually would suggest" (p. 358). Nevertheless, it is useful to organize ideas in classified guidelines for communication purposes.
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Table 1. Transition from concrete to abstract Ann and Tom together have $6. Tom has twice amount of money than Ann. How much money has each? Verbal representation Diagram Symbol The amount of money Ann has x unknown The amount of money Tom has -- -- unknown + unknown or 2x 2 unknown Total amount of money unknown + unknown + unknown -- -- -- x + 2 x = 3x or 3 unknown This representation shows that the $6 x + 2 x = 6 total is $6 -- -- -- 3 x = 6 3 unknown = $6 unknown = $2 or Ann has $2 --2 x = 2 2unknown = $4 or Tom has $4 -- --$4 2x = 4 Table 2. The taxonomy for assessing conceptual understanding in algebra Phases Description Examples Phase 0 The students Given the equation, e.g., 2x + 3 = 9, the students No signs of i) are not able to cope use instantiations and conceptual with the level of trial and error method understanding abstraction embedded into exclusively; they do not the problem, thus show treat the equation as an the persistent need to algebraic entity, and reduce the algebraic struggle to dealing with problem/concept to the unknown quantities lower level; presented via symbols and diagrams (e.g., line ii) prefer numerical segment image). They need instantiations; to think about unknown as a real life object (e.g., iii) prefer memorizing reduce the equation 2x=9 the rules and procedures; to the equation 2unknown = 9 or 2dollars = 9) to iv) do not recognize be able to manipulate the structura0y the same quantities. These concept/relationship students can not describe posed via different their observations in representational adequate terms due to modalities. limited vocabulary. They struggle with operating upon mathematical laws, properties and conventions. They might prefer to "thinking in pictures", but due to the lack of diagrammatic literacy they are likely to become confused with their own pictorial representations of the concerts. Phase 1 The students Given an equation presented in symbols, the Transitional v) cope with the level of students use algebraic phase abstraction embedded into methods to find the the problems, i.e., are solution (i.e., isolate able to solve the the unknown, use the problems with unknown inverse operations). They quantities without using are also able to solve instantiations; problems involving linear relationship with one vi) comfortable to 'think unknown posed in words in symbols' and diagrams. Their sense of symbol might be still vii) remember rules and limited to the objects procedures; that the symbols represent (e.g., some viii) show relatively might need to think of 2s fluent procedural skills + 3c as 2sodas + when solving equations 3cookies). When presented posed in symbolic torn; with different modes of structurally the same ix) may not produce relationship, they can algebraic equations that only partially describe would describe the some similarities (e.g., relationship presented in the problems have the words and/or diagrams; same solution; the problems use the same x) might observe some numbers). Undeveloped similarities between mathematics language/ different representations vocabulary might create a of structurally the same barrier for them to be relationship but do not able to express their fully recognize the same thinking in adequate relationship posed via terms. The lack of multiple representations. mathematics language fluency prevents them from flexible operation upon multiple representations. Phase 2 The students The students are able to explain the nature of the Conceptual xi) cope with the level concept and its understanding of abstraction presented properties using correct phase by the concept and/or terminology. They problem; xii) demonstrate demonstrate fluency in flexible thinking the sophisticated language of algebra by xiii) are able to explain confident use of its the full meaning of the vocabulary and meanings. concepts of unknown, They have developed sense linear relationships and of symbol (i.e., need not properties of the linear to reduce symbols to the equations with one material objects, and unknown (e.g., treat the symbols as reflection, symmetry, mathematical entities). equivalence). Given the equation, e.g., The most critical 3 X n = 15, the students indicators of the correctly produce a conceptual understanding sophisticated diagrammatic xiv) demonstration of the representation that ability to recognize adequately describes the structurally the same relationship: relationship; Given a word problem and/ xv) being able to or a diagram, the fluently translate and students produce an interpret different algebraic equation that representations of adequately models the structurally the same relationship. relationship.
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|Author:||Panasuk, Regina M.|
|Publication:||College Student Journal|
|Date:||Jun 1, 2011|
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