• Schoenfeld and Arcavi (1988) argue that “understanding the concept [of a variable] provides the basis for the transition from arithmetic to algebra and is necessary for the meaningful use of all advanced mathematics.” Yet, the concept of a variable is “more sophisticated” than teachers expect and it frequently becomes a barrier to a student’s understanding of algebraic ideas (Leitzel, 1989). For example, some students have a difficult time shifting from a superficial use of “a” to represent apples to a mnemonic use of “a” to stand for the number of apples (Wagner and Kieren, 1989).
• Students treat variables or letters as symbolic replacements for specific unique numbers. As a result, students expect that x and y cannot both be 2 in the equation x+y=4 or that the expression “x+y+z” could never have the same value as the expression “x+p+z” (Booth, 1988).
• Students have difficulty representing and solving algebraic word problems because they rely on a direct syntax approach which involves a “phrase-by-phrase” translation of the problem into a variable equation (Chaiklin, 1989; Hinsley et al., 1977). An example of this difficulty is the common reversal error associated
with the famous “Students-and-Professors” problem: Write an equation using the variables S and P to represent the following statement: “There are 6 times as many students as professors at this university.” Use S for the number of students and P for the number of professors. A significant number of adults and students (especially engineering freshmen at MIT) write the reversal “6S=P” instead of the correct expression “S=6P.” Clement et al. (1981) suggest that the reversal error is prompted by the literal translation of symbols to words, where S is read as “students” and P as “professors” rather than S as “the number of students” and P as “the number of professors.” Under this interpretation, the phrase “6 students are equal to 1 professor” becomes a ratio.
• Students often can describe a procedure verbally yet not be able to recognize the algebraic representation of this same procedure (Booth, 1984).
• Students try to force algebraic expressions into equalities by adding “=0” when asked to simplify or evaluate (Wagner et al., 1984; Kieren, 1983).
• The concept of a function is the “single most important” concept in mathematics education at all grade levels (Harel and Dubinsky, 1992).
• Students have trouble with the language of functions (e.g., image, domain, range, pre-image, one-to-one, onto) which subsequently impacts their abilities to work with graphical representations of functions (Markovits et al., 1988).
• Students tend to think every function is linear because of its early predominance in most algebra curricula (Markovits et al., 1988). The implication is that nonlinear functions need to be integrated throughout the students’ experience with algebra.
• Students, surrounded initially by function prototypes that are quite regular, have cognitive difficulties accepting the constant function, disconnected graphs, or piece-wise defined functions as actually being functions (Markovits et al., 1988).
• In Dreyfus’ (1990) summary of the research on students’ working to understanding functions, three problem areas are identified:
1. The mental concept that guides a student when working with a function in a problem tends to differ from both the student’s personal definition of a function and the mathematical definition of a function.
2. Students have trouble graphically visualizing attributes of a function and interpreting information represented by a graph.
3. Most students are unable to overcome viewing a function as a procedural rule, with few able to reach the level of working with it as a mathematical object.
• Students’ transition into algebra can be made less difficult if their elementary curriculum includes experiences with algebraic reasoning problems that stress representation, balance, variable, proportionality, function, and inductive/deductive reasoning (Greenes and Findell, 1999).
• Students may be able to solve traditional problems using both algebraic and graphical representations, yet have minimal understanding of the relationships between the two representations (Dreyfus and Eisenberg, 1987; DuFour et al., 1987).
• Graphing technologies encourage students to experiment with mathematics, sometimes with negative effects. In an algebra or precalculus context, visual llusions can arise that actually are student misinterpretations of what they see in a function’s graphical representation. For example, students view vertical shifts as horizontal shifts when comparing linear graphs (such as the graphs of y=2x+3 and y=2x+5). Also, students falsely conclude that all parabolas are not similar due to the misleading effects of scaling. Students often conclude that a function’s domain is bounded due to misinterpretations of the graphing window (Goldenberg, 1988).
• Students have more facility working with functions represented graphically than functions represented algebraically. The graphical representations seem to visually encapsulate the domain, range, informal rule, and behavior of the function in a manner that the algebraic form cannot (Markovits et al., 1988). In turn, highability students prefer using the graphical representation, while low-ability students prefer a tabular representation of the function (Dreyfus and Eisenberg, 1981).
• Students misinterpret time-distance graphs because they confuse the graph with the assumed shape of the road. Also, students do not necessarily find it easier to interpret graphs representing real-world contexts when compared to graphs representing “symbolic, decontextualized” equations (Kerslake, 1977).
• Students have a difficult time interpreting graphs, especially distance-time graphs. Intuitions seem to override their understandings, prompting students to view the graph as the path of an actual “journey that was up and down hill” (Kerslake, 1981).
• Students have difficulty accepting the fact that there are more points on a graphed line than the points they plotted using coordinates. This is known as the continuous vs. discrete misconception. Some students even contend that no points exist on the line between two plotted points, while other students accept only one possible such point, namely the mid-point (Kerslake, 1981).
• Middle school students find constructing Cartesian graphs difficult, especially with regard to their choice of a proper scaling, positioning the axes, and understanding the structure involved (Leinhardt et al., 1990).
• Precalculus students’ use of graphing calculators improved their understanding of the connections between a graph and its algebraic representation (in contrast to students learning the same content without calculators). Also, the calculator-using students tended to view graphs more globally (i.e., with respect to continuity, asymptotic behavior, and direction changes) and showed a better understanding of the underlying construction of graphs, especially the use of scale (Rich, 1990).
• The oversimplified concept of slope taught to students in an algebra class can lead to misconceptions when working with the concept of slope as a part of differentiation in a calculus class (Orton, 1983).
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