Errors with Algebra
Booth and colleagues (Booth, Barbieri, Eyer, & Pare-Blagoev, 2014) analyzed the errors of Algebra I students across five school districts in four states, in order to understand not only which errors were most common but also which ones were most detrimental to algebra achievement as measured by standardized test items. Based on the extant literature, they coded for conceptual errors with variables, fractions, negatives, operations, mathematical properties, and equality/inequality. For comparison, they also coded for errors with arithmetic.
Across the year, the most prevalent errors were those involving negatives, arithmetic, variables, and the equal sign. A more nuanced analysis revealed a time sensitive aspect for the contribution of particular errors to the end of year assessment scores. For example, while negative number errors were prevalent throughout the year, making those errors at the end of the year was predictive of lower scores on the end of year assessment items. A similar result was found for arithmetic. Errors with equal/ inequality became increasingly prevalent throughout the year, and making these errors in the middle or end of the year was predictive of lower end of year scores. Making errors with variables, such as combining unlike terms, in the first part of the year also predicted lower end of year scores.
Some errors that were not among the most common ones were still predictive of end of year scores. In particular, students who began the year making errors with operations (e.g., treating 5 + x as 5x) and with properties (e.g., treating 5 - x as x - 5) were more likely to have lower end of year algebra scores.
The IES practice guide for improving algebra knowledge outlines three researchbased recommendations (Star et al., 2015). The first recommendation is to use solved problems, also known as worked examples, to help students analyze and reason about algebraic strategies. The second recommendation is to help students to notice and make use of underlying structures, including the presence and placement of variables and symbols, in algebra problems so that they can see similarities across problem types. The final recommendation is for students to learn and choose from multiple strategies when solving algebra problems. These first two, worked examples and noticing structure, are addressed in this section. The recommendation to learn and use multiple strategies is addressed in the following section, which is focused on procedural flexibility.
Worked Examples. The use of worked examples, which includes a problem presented along with solution method, has a long history in the cognitive and educational psychology literature, particularly for learning mathematics (Atkinson, Derry, Renkl, & Wortham, 2000). Research generally supports the use of worked examples early in the skill acquisition process, along with prompts for selfexplanations so that the learner is asked to elaborate on what is presented in the example. For optimal learning, worked examples are interleaved with problems for the learners to solve.
Worked examples with self-explanation prompts have been used along with practice problems to effectively support learning in algebra classrooms. Incorrect examples may be especially important for supporting conceptual knowledge of algebra (Booth, Lange, Koedinger, & Newton, 2013), given they are designed to target common errors/misconceptions. Students with low prior knowledge seem to benefit most from either incorrect worked examples (Barbieri & Booth, 2016) or a combination of correct and incorrect examples (Booth et al., 2015). In a recent study, Barbieri, Miller-Cotto, and Booth (2019) found that eighth graders prone to making conceptual errors as they practiced graphing-related skills benefited more from studying correct and incorrect examples with visual signaling cues than from worked examples with self-explanations. These visual cues were designed to signal important conceptual features in the problems, such as identifying slope and y-intercept within the worked example. Students who made fewer errors during practice problems benefited similarly from worked examples with self-explanations and with visual signaling cues.
Noticing and Using Structure. To mathematicians, noticing and using structure is a fundamental aspect of doing mathematics, so fundamental that mathematics can be described as “the science of structure” (Newton & Sword, 2018, p. 33). In algebra, structure can be thought of as “the underlying mathematical features and relationships of an expression, representation, or equation” (Star et al., 2015, p. 6). As noted above, feature knowledge such as knowledge of the equal sign, negatives, and variables is critical to success in algebra. Noticing structure means noticing how these features are arranged in an algebraic representation and being able to see meaning in those arrangements. For example, a student who notices structure can clearly understand that 2+3 + x = 10 and 2+3 = x + 10 have quite different meanings, despite the surface similarities.
Experts including mathematicians assert that using precise language can support an understanding of structure (Newton & Sword, 2018; National Governors Association Center for Best Practices & the Council of Chief State School Officers, 2010; Star et al., 2015). For example, they caution against imprecise language such as stating, “two negatives make a positive,” because although a negative times a negative results in a positive number, a negative plus a negative does not. Students sometimes overgeneralize these kinds of imprecise statements, making errors such as thinking -5a - 4a is 9a (Vlassis, 2004).
One way to capture students’ noticing of structure is to measure errors while encoding equations (McNeil & Alibali, 2004). Booth and Davenport (2013) measured feature encoding of algebra by displaying an algebraic equation on a screen for 5 seconds and then asking students to recreate the equation on their paper after it disappears from the screen. Their findings confirm that feature knowledge of algebra supports feature encoding and predicts the ability to solve algebraic equations.
Varying the Format. The equal sign is one feature that has received significant attention in the literature. As noted in the example above, the placement of the equal sign matters. Students who view the equal sign as an indication to find an answer may just add all of the constants together no matter their placement in the equation (e.g., erroneously finding 15 as the value of x for 2+3 = x + 10). In a study by McNeil and Alibali (2004), fourth graders making this error frequently encoded non-conventional equations as if they were conventional ones, with all to-be-added terms on the lefthand side of the equal sign (e.g., encoding 4 + 5 = 2 +__as 4 + 5 + 2 =__). Although in
late elementary school most students have developed a basic relational understanding of the equal sign, where they recognize it as an indication that two expressions have the same value, a robust understanding of equivalence continues to develop through middle school (Fyfe, Matthews, Amsel, McEldoon, & McNeil, 2018; Rittle-Johnson, Matthews, Taylor, & McEldoon, 2011).
A simple, yet effective intervention for addressing equal sign errors is to more frequently pose unconventional arithmetic problems so that students are not overly accustomed to all terms being on the left of the equal sign, with only a blank for the answer on the right. Under the CCSS, students are expected to be presented with simple, unconventional problems as early as first grade (National Governors Association Center for Best Practices & the Council of Chief State School Officers, 2010). McNeil, Fyfe, and Dunwiddie (2015) provided this kind of modified practice as part of an intervention with second graders. Modified practice workbooks in their study included unconventional problems (e.g.,__= 3 + 5 and 8 =__+ 5), whereas students in a control group received conventional problems to practice (e.g., 3 + 5 =__). To support a
relational understanding of equivalence, the equal sign was sometimes replaced with words that conveyed its meaning, such as “is the same amount as”. Finally, the problems were organized such that students encountered several in a row with the same sum (e.g.,__= 3 + 5 and__= 2 + 6). Students in the modified workbook condition
outperformed students in the control, with lasting effects.
Although the modified practice workbooks (McNeil et al., 2015) resulted in positive, long-term effects compared to traditional practice, not all students achieved a robust understanding of the equal sign. Building on that work, McNeil and colleagues designed and tested a more comprehensive intervention for targeting equal sign knowledge (McNeil, Hornburg, Brletic-Shipley, & Matthews, 2019). In their study of 142 second grade students, an active control group was compared to a comprehensive intervention group. The active control group received practice problems similar to the intervention in McNeil et al. (2015). The comprehensive intervention group included three additional elements: (1) introduction of the equal sign in non-arithmetic situations (e.g., 5 = 5); (2) concreteness fading, where representations became more abstract over time; and (3) prompts for comparing problem formats and strategies (including incorrect ones). Students randomly assigned to the comprehensive intervention improved significantly more in their understanding of equivalence than students assigned to the active control classrooms, with large effect.
Given that students are introduced to the equal sign very early, much of the research on this topic has been conducted with elementary school children. However, as noted by Booth et al. (2014), students taking algebra make errors related to equal sign knowledge, and these errors predict lower end of year algebra scores. Therefore, more research is needed on how to remediate this knowledge for students learning algebra.
Representations. Asking students to discuss, compare, and move between different algebraic representations, such as equations, graphs, word problems, and diagrams, can support student understanding of underlying algebraic structures (National Council of Teachers of Mathematics, 2000; Star et al., 2015). Additionally, some researchers and educators recommend physical models to support student learning. A balance scale, for example, provides students with a model for understanding that the equal sign indicates that two expressions have the same value. Related, it also provides insight into the idea that the same operation is performed to both sides in order to maintain that equivalency. In a study with 40 eighth graders, Vlassis (2002) found that the model was limited, however, in its ability to help students understand negatives within the equation.
As with fractions, number lines are one way to represent negatives. Tsang, Blair, Bofferding, and Schwartz (2015) compared three instructional conditions for fourth graders learning about negatives, and all three conditions incorporated the number line as the primary representation. The Jumping condition included a small figure that moved along the number line in a way that corresponded to an addition problem (e.g., for 5 + -3, starting at 5 and “jumping” to the left three spaces). The Stacking condition included magnetic manipulatives with different colors to represent positives and negatives. For 5 + -3, five blue blocks are placed on the number line, between 0 and 5. Three red blocks are then stacked on top of the blue ones, starting at 5. Each red and blue pair “cancel” each other, so two blue blocks are left. The Folding condition involved place the 5 blue blocks between 0 and 5, and the 3 red blocks between 0 and -3. Then, the blocks were folded toward each other, with three of them “canceling” each other out. The physical movement of folding in this condition was designed to emphasize the underlying symmetrical structure inherent in the number line. Students in the Folding condition outperformed students in the other two groups on items not yet taught, such as negative fractions and algebra readiness problems (e.g., 2 + -3 =__+ -1).
Comparing. The use of worked examples as an effective instructional tool is not limited to single examples paired with prompts for self-explanation. Another approach supported by research is to use two worked examples, presented side-by-side, in order help students to make comparisons and draw important conclusions. In a study that incorporated pairs of side-by-side worked examples into year-long algebra classrooms, greater use of the comparison materials was associated with greater gains in procedural knowledge (Star et al., 2015). These materials included four types of comparisons, often involving one problem worked in two ways. For example, a correct method of solving a problem might be compared to an incorrect method. Alternatively, one method might highlight why the other one works. Or, a general and alternative method might be compared in order to promote flexible problem solving. Other times, two different problems might be compared. For example, two variations of a problem might be compared in order to highlight a particular idea (e.g., slope), or two visually similar problems might be compared in order to highlight an important difference (e.g., x3-x4 and (x3)4).
Ziegler and Stern (2016) studied this last type of comparison in particular and found long-term benefits for sixth graders with little or no prior knowledge of algebra. Rather than studying worked out examples, these students were exposed to direct instruction as a means to compare and contrast problems with similar features (e.g., 3x + 3x + 3x and 3x-3x-3x~). The contrast group was presented with addition and multiplication problems simultaneously, whereas the sequential group was presented with addition problems followed by multiplication problems. Although the sequential group outperformed the contrast group on immediate learning, the contrast group outperformed the sequential group on follow-up measures at three time points.
Comparison of this kind holds promise for alleviating some of the errors identified by Booth et al. (2014). Specifically, comparing expressions with similar features but different operations (e.g., 5 + x and 5x) or different ordering of terms (e.g., 5 - x and x - 5) may help students understand and remember important ideas that are critical for success in algebra. It may also be helpful for addressing errors with negative numbers. During interviews with students about integer arithmetic, Bishop, Lamb, Philipp, Whitacre, and Schappelle (2016) found that some students naturally used these kinds of comparisons when they explained their reasoning about integers. In their comparisons, students varied the sign of the numbers (e.g., 6-2 and 6 - (-2)), the operations (e.g., -8 + 3 and -8 - 3), and features such as the order of the addends (e.g., -5 + 2 and 5 + -2) in order to logically reason about what must be true or not true for a given problem. For example, a student reasoned that 6 - (-2) cannot be 4 because 6 - 2 is 4.
Although these kinds of comparisons were not prevalent in the Bishop et al. (2016) study, they occurred across a variety of grade levels. The researchers therefore suggested that it may be fruitful to purposefully incorporate comparisons of this sort into instruction, along with prompts to support students’ reasoning about them. They also suggested that students have more opportunities to make conjectures about how operations work, to encourage students to think more about the underlying mathematical structures. One teacher who regularly did this found her students were able to correctly conjecture that dividing across numerators and denominators is a valid approach to fraction division (Newton et al., 2014), a fact that is often obscured through traditional instruction. Systematic research is needed to understand the effects of this kind of intervention.
As with fraction knowledge, knowledge of algebra requires sense-making. Students need to understand algebraic symbols and the meanings that are conveyed by them and the way they are organized. These understandings can support deep and accurate strategy use in algebra. The IES practice guide for algebra learning recommends the use of worked examples to help students understand algebraic strategies.
Incorrect worked examples can be especially helpful for targeting errors based on common misconceptions. A second recommendation includes helping students notice and use structure. Similar to fractions, representations such as number lines, concrete manipulatives, and word problems, can help. Graphs and equations are also critically important representations for understanding algebraic structures. Varying the format of the equations can highlight important structures such as the equal sign. Comparison also supports noticing and using structure, as it can highlight important similarities or differences between representations that might otherwise be obscured.