Errors with Fractions
The challenge posed by fractions, along with the important role of fractions for learning algebra, has resulted in increased attention for this topic (Siegler et al., 2013). Because fractions do not have all of the same properties as whole numbers, children must adjust their thinking to accommodate this “new” kind of number. If they do not, it may lead to errors based on misconceptions about fractions, even as they try to integrate information from instruction (Siegler, Thompson, & Schneider, 2011; Stafylidou & Vosniadou, 2004).
Knowledge of fraction magnitudes, or relative sizes of fractions, is especially critical for later mathematics learning. It predicts overall mathematics achievement as well as learning in algebra (Booth, Newton, & Twiss-Garrity, 2014; Siegler et al., 2011). In their study of 200 students, ages 10-16, Stafylidou and Vosniadou (2004) analyzed responses and justifications to ordering tasks and identified different categories of misconceptions students held as they learned about fractions. For example, students may believe that 4A is smaller than % because 3 and 4 are smaller than 5 and 6. In this case, they are treating the numbers as if they are all whole numbers. As students learn about the size of fractional parts relative to the denominator, they may erroneously claim that % is smaller than % because the larger numbers indicate smaller fractions. Fazio et al. (2016) found that low performing students at a community college used the “bigger denominator” strategy frequently for comparing fractions, resulting in many errors. On the other hand, Malone and Fuchs (2017) found the whole number error to be especially pervasive among a group of at-risk fourth graders.
Students also make a variety of errors with fraction arithmetic, often following predictable patterns (Braithwaite, Pyke, & Siegler, 2017; Newton, Willard, & Teufel, 2014; Siegler & Pyke, 2013). As with comparing fractions, they make errors by overgeneral-izing their knowledge of whole numbers. A well-documented example is treating fraction addition as two separate whole number addition problems, erroneously adding across numerators and denominators (e.g., '/2 + % = ¥5). Students also make errors by overgeneralizing rules for fraction arithmetic. A student using the incorrect strategy above might instead be thinking of fraction multiplication, for which you multiply across numerators and denominators. Another common error in this category would be to find (or keep) a common denominator when it is not appropriate to do so, such as with fraction multiplication (e.g., 2/s x ¥s = %) or fraction division (e.g., % -r ¥s = ¥5). Students making this error have most likely learned how to add fractions, in which case a common denominator is needed.
The Institute of Education Sciences (IES) practice guide (Siegler et al., 2010) for effective fraction instruction offers several recommendations for targeting the erroneous strategies described above. The two recommendations with moderate (as opposed to minimal) research-based evidence are as follows. First, students need to “recognize that fractions are numbers and that they expand the number system beyond whole numbers” (p. 19). Interventions that make use of number lines in particular are recommended, given they can assist students in understanding the place of fractions within the number system (Siegler et al., 2010; Wu, 2009). The Common Core State Standards for mathematics recommends introducing fractions using number lines to students as early as third grade (NGA Center & CCSSO, 2010). Second, students need to “understand why procedures for computations make sense” (Siegler et al., 2010, p. 26).
Number Lines. Wu (2009) posited that making use of number lines as a teaching tool provides coherence to the study of numbers. “In particular, regardless of whether a number is a whole number, a fraction, a rational number, or an irrational number, it takes up its natural place on this line” (p. 8). Number lines can support students’ understanding of magnitude and equivalence, as well as computations with fractions. Unfortunately, research focused on number line interventions has been scarce. Shin and Bryant (2015) synthesized the fraction intervention research that targeted students struggling to learn mathematics. Across 17 studies, most interventions included concrete and visual representations. However, none of them included number lines.
In a recent study, Hamdan and Gunderson (2017) found a causal effect for number line training. Second and third grade students learning about fractions using a number line out-performed students who were introduced to fractions with an area model on a transfer task that involved fraction comparison. Only the students using a number line representation were able to use what they had learned to compare two fractions.
Fuchs and colleagues (Fuchs, Malone, Schumacher, Namkung, & Wang, 2017) recently reported on the impact of a multi-year program to target at-risk students’ fraction knowledge by emphasizing fraction magnitude, including the use of number lines. In a series of studies, they compared performance for students involved in the intervention to a business-as-usual condition, where the primary focus was on a part-whole interpretation of fractions. Students in the intervention group outperformed the control each year on the number line estimation task, which asked students to place fractions and mixed numbers on a number line marked 0 and 2 at the endpoints. Students in the intervention group also outperformed control on addition and subtraction of fractions and mixed numbers. Finally, the intervention group outperformed control on a set of released items from the National Assessment of Educational Progress (NAEP) that emphasized magnitude and part-whole equally.
Over several years of the project, Fuchs and colleagues (Fuchs et al., 2017) tested the effect of additional supports beyond the emphasis on magnitude. For example, in year 2 they varied the type of practice students experienced. One group was focused on fluency, while the other was focused on explaining their thinking. Although no significant difference was found for the three fraction measures, they did find a moderating effect for working memory. In particular, students with very low working memory benefited more from the explaining condition, while students with high working memory benefited more from the fluency practice. However, given that a larger proportion of students benefited from fluency practice, this form of practice was included in the program for subsequent years.
In year 3, additive and multiplicative word problem conditions were included using schema-based instruction, where students were taught to recognize the problem type and to represent the type with a number sentence or visual display. A control group focused on identifying key words in the word problems. The multiplicative group outperformed the additive and control groups on multiplicative word problems and outperformed the control group on additive word problems. The additive group outperformed the other two groups on additive problems but performed similarly to the control group on multiplicative word problems (Fuchs et al., 2017).
In year 4, the researchers compared the added effect of supported self-explanations compared to training in solving multiplicative word problems (Fuchs et al., 2017, 2016). Rather than requiring at-risk students to produce their own explanations, self-explanations were modeled and the students analyzed, discussed, and elaborated on them. On word problems, students in the word problem group tended to outperform the explanation group. However, for both magnitude comparison and quality of explanations, students in the supported self-explanation group tended to outperform the word problem group. Moreover, students with both high and low working memory performed similarly in this condition.
The work by Fuchs and colleagues (Fuchs et al., 2017, 2016) described above underscores the importance of emphasizing fraction magnitudes, including the use of number lines, to promote early fraction knowledge. As noted by the authors, their program targeted fourth grade standards, and so additional research is needed “to address the challenges associated with multiplying and dividing fractions as well as other complex mathematics curricular targets” (p. 638).
Understanding Why Procedures Work. Mathematics educators have long held an interest in promoting deep understanding of fraction procedures, suggesting that simply knowing how to compute with fractions is insufficient in many ways. Highlighting this distinction, Skemp (1976) referred to knowing “how” as having an instrumental understanding, while knowing both “how and why” suggests a relational understanding. For adding fractions, this distinction would suggest, on the one hand, knowing that you need to find common denominators to add V2 and Vs versus, on the other, knowing that you need to find common denominators so that the fractions are renamed using same-sized parts, making it easy to find out the total number of those parts.
More broadly, Hiebert and Lefevre (1986) characterized conceptual knowledge “as knowledge that is rich in relationships” (p. 3). This characterization of knowledge could include the latter example above, as well as other important relationships. For example, you might know that V2 and Vs can also be renamed as decimals and then added, and that the sum is equivalent to the one obtained using fractions. You might also be able to estimate the sum, knowing it must be slightly larger than V2, the first addend. Unfortunately, even when students can accurately identify which fraction is the larger one or estimate the magnitude of each fraction, many are unsuccessful at estimating the sum of the two fractions (Braithwaite, Tian, & Siegler, 2018; Cramer & Wyberg, 2007). This lack of understanding, as well as errors such as the overgeneralizations described above, has prompted researchers and educators to search for ways to support students in making sense of fraction computation.
Visual Representations. As indicted above, the number line is an important representation for understanding fractions, and Fuchs and colleagues (2017) have reported good success at integrating this representation into a program designed to strengthen childrens understanding of fraction magnitudes. Research is needed to extend this work to fraction computation. On the other hand, extensive research has focused on concrete manipulatives for visually representing fractions, in order to help students make sense of them. Research on the use of manipulatives in mathematics and science generally supports fading from concrete to abstract representations, as suggested by Bruner’s modes of representation (Fyfe, McNeil, Son, & Goldstone, 2014). According to Bruner, the use of concrete and pictorial models should precede work with symbols only. Students need support linking these representations, especially the pictorial and symbolic ones. A meta-analysis focused on fraction skills found that a concrete to abstract progression showed promise for students with disabilities, although more research is needed (Ennis & Losinksi, 2019).
Word Problems. Building on extensive research by Carpenter and colleagues that focused on childrens strategies for solving whole number problems (Carpenter et al., 1996), Empson and Levy (2011) emphasized word problems as a critical tool for sense-making with fractions. They recommended a progression of problem types with instructional support to help formalize student thinking about these problems. For example, when asked an equal sharing problem about four children sharing three cookies, a student may draw a picture of a cookie, split it into four parts, and distribute each part to a person. Repeating this process for each cookie results in three one-fourths for each person. In response to this strategy, a teacher might provide a number sentence to match the drawing, such as Vi + ’A + Vi = %. These researchers emphasized that a key aspect of this approach involves making mathematical ideas and properties explicit by linking students’ strategies directly to the mathematical sentences that represent them.
There is a long history of educators and psychologists promoting sense-making in mathematics. According to the IES practice guide for effective fraction instruction (Siegler et al., 2010), two general areas of sense-making for fractions are well-supported by research. First, students need to understand that fractions are numbers. Second, students need to understand why fraction procedures work the way they do. Without these understandings, students tend to overgeneralize their prior knowledge in ways that are not appropriate. Interventions targeting these errors highlight representations as key to fostering understanding and sense-making. Representing fractions on the number line, with concrete manipulatives and within word problems are all important for fostering deep understanding and appropriate strategy use.