Theses Doctoral

# Recharging Rational Number Understanding

Schiller, Lauren Kelly

In 1978, only 24% of 8th grade students in the United States correctly answered whether 12/13+7/8 was closest to 1, 2, 19, or 21 (Carpenter, Corbitt, Kepner, Lindquist, & Reys, 1980). In 2014, only 27% of 8th grade students selected the correct answer to the same problem, despite the ensuing forty years of effort to improve students’ conceptual understanding (Lortie-Forgues, Tian, & Siegler, 2015). This is troubling, given that 5th grade students’ fraction knowledge predicts mathematics achievement in secondary school (Siegler et al, 2012) and that achievement in math is linked to greater life outcomes (Murnane, Willett, & Levy, 1995). General rational number knowledge (fractions, decimals, percentages) has proven problematic for both children and adults in the U.S. (Siegler & Lortie-Forgues, 2017). Though there is debate about which type of rational number instruction should occur first, it seems it would be beneficial to use an integrated approach to numerical development consisting of all rational numbers (Siegler, Thompson, & Schneider, 2011). Despite numerous studies on specific types of rational numbers, there is limited information about how students translate one rational number notation to another (Tian & Siegler, 2018).

The present study seeks to investigate middle school students’ understanding of the relations among fraction, decimal, and percent notations and the influence of a daily, brief numerical magnitude translation intervention on fraction arithmetic estimation. Specifically, it explores the benefits of Simultaneous presentation of fraction, decimal, and percent equivalencies on number lines versus Sequential presentation of fractions, decimals, and percentages on number lines. It further explores whether rational number review using either Simultaneous or Sequential representation of numerical magnitude is more beneficial for improving fraction arithmetic estimation than Rote practice with fraction arithmetic. Finally, it seeks to make a scholarly contribution to the field in an attempt to understand students’ conceptions of the relations among fractions, decimals, and percentages as predictors of estimation ability.

Chapter 1 outlines the background that motivates this dissertation and the theories of numerical development that provide the framework for this dissertation. In particular, many middle school students exhibit difficulties connecting magnitude and space with rational numbers, resulting in implausible errors (e.g., 12/13+7/8=1, 19, or 21, 87% of 10>10, 6+0.32=0.38). An integrated approach to numerical development suggests students’ difficulty in rational number understanding stems from how students incorporate rational numbers into their numerical development (Siegler, Thompson, & Schneider, 2011). In this view, students must make accommodations in their whole number schemes when encountering fractions, such that they appropriately incorporate fractions into their mental number line. Thus, Chapter 1 highlights number line interventions that have proven helpful for improving understanding of fractions, decimals, and percentages.

In Chapter 2, I hypothesize that current instructional practices leave middle school students with limited understanding of the relations among rational numbers and promote impulsive calculation, the act of taking action with digits without considering the magnitudes before or after calculation. Students who impulsively calculate are more likely to render implausible answers on problems such as estimating 12/13+7/8 as they do not think about the magnitudes (12/13 is about equal to one and 7/8 is about equal to one) before deciding on a calculation strategy, and they do not stop to judge the reasonableness of an answer relative to an estimate after performing the calculation. I hypothesize that impulsive calculation likely stems from separate, sequential instructional approaches that do not provide students with the appropriate desirable difficulties (Bjork & Bjork, 2011) to solidify their understanding of individual notations and their relations.

Additionally, in Chapter 2, I hypothesize that many middle school students are unable to view equivalent rational numbers as being equivalent. This hypothesis is based on the documented tendency of many students to focus on the operational rather than relational view of equivalence (McNeil et al., 2006). In other words, students typically focus on the equal sign as signal to perform an operation and provide an answer (e.g., 3+4=7) rather than the equal sign as a relational indicator (e.g., 3+4=2+5). Moreover, this hypothesis is based on the documented whole number bias exhibited by over a quarter of students in 8th grade, such that students perceived equivalent fractions with larger parts as larger than those with smaller parts (Braithwaite & Siegler, 2018b). If middle school students are unable to perceive equivalent values within the same notation as equivalent in size, it seems probable that they might also struggle perceiving equivalent rational numbers as equivalent across notations. This is especially true in light of evidence that many teachers often do not use equal signs to describe equivalent values expressed as fractions, decimals, and percentages (Muzheve & Capraro, 2012). Chapter 2 underscores the importance of highlighting the connections among notations by discussing the pivotal role of notation connections in prior research (Moss & Case, 1999) and the benefit of interleaved practice in math (Rohrer & Taylor, 2007). Finally, I propose a plan for improving students’ understanding of rational numbers through linking notations with number line instruction, as an integrated theory of numerical development (Siegler et al, 2011) suggests that all rational numbers are incorporated into one’s mental number line.

Chapter 3 details two experiments that yielded empirical evidence consistent with the hypotheses that students do not perceive equivalent rational numbers as equivalent in size and that this lack of integrated number sense influences estimation ability. The findings identify a discrepancy in performance in magnitude comparison across different rational number notations, in which students were more accurate when presented with problems where percentages were larger than fractions and decimals than when they were presented with problems where percentages were smaller than fractions and decimals. Superficially, this finding of a percentages-are-larger bias suggests students have a bias towards perceiving percentages as larger than fractions and decimals; however, it appears this interpretation is not true on all tasks. If students always perceive percentages as larger than fractions and decimals, then their placement of percentages on the number line should be larger than the equivalent fractions or decimals. However, this was not the case. The experiments revealed that students’ number line estimation was most accurate for percentages rather than the equivalent fraction and decimal values, demonstrating that students who are influenced by the percentages-are-larger bias are most likely not integrating understanding of fractions, decimals, and percentages on a single mental number line. Furthermore, empirical evidence provided support for the theory of impulsive calculation defined earlier, such that many students perform worse when presented with distracting information (“lures”) meant to elicit the use of flawed calculation strategies than in situations without such lures. Importantly, integrated number sense, as measured by the composite score of all cross-notation magnitude comparison trials, was shown to be an important predictor of estimation ability in the presence of distracting information on number lines and fraction arithmetic estimation tasks, often above and beyond number line estimation ability and general math ability.

The experiments reported in Chapter 3 also evaluated whether Simultaneous, integrated instruction of all notations improved integration of rational number notations more than Sequential instruction of the three notations or a control condition with Rote practice in fraction arithmetic. The experiments also evaluated whether the instructional condition influenced fraction arithmetic estimation ability. The findings supported the hypothesis that a Simultaneous approach to reviewing rational numbers provides greater benefit for improving integrated number sense, as measured by more improvement in the composite score of magnitude comparison across notations. However, there was no difference among conditions in fraction arithmetic estimation ability at posttest. The experiments point to potential areas for improvement in future work, which are described subsequently.

Chapter 4 attempts to explore further students’ understanding of the relations among notations. For this analysis, a number of data sources were examined, including student performance on assessments, interview data, analysis of student work, and classroom observations. Three themes emerged: (1) students are employing a flawed translation strategy, where students concatenate digits from the numerator and denominator to translate the fraction to a decimal such that a/b=0.ab (e.g., 3/5=0.35). (2) percentages can serve as a useful tool for students to judge magnitude, and (3) students equate math with calculation rather than estimation (e.g., in response to being asked to estimate addition of fractions answers, a student responded, “I can’t do math, right?”). Moreover, case studies investigated the differential effect of condition (Simultaneous, Sequential, or Control) on students’ strategy use. The findings suggest that the Simultaneous approach facilitated a more developed schema for magnitude, which is crucial given that a student’s degree of mathematical understanding is determined by the strength and accuracy of connections among related concepts (Hiebert & Carpenter, 1992).

Chapter 5 concludes the dissertation by discussing the contributions of this work, avenues for future research, and educational implications. Ultimately, this dissertation advances the field of numerical cognition in three important ways: (1) by documenting a newly discovered bias of middle school students perceiving percentages as larger than fractions and decimals in magnitude comparisons across notations and positing that a lack of integrating notations on the same mental number line is a likely mechanism for this bias; (2) by demonstrating that students exhibit impulsive calculation, as measured by the difference in performance between situations where students are presented with distracting information (“lures”) meant to elicit the use of flawed calculation strategies and situations that do not involve lures; and (3) by finding that integrated number sense, as measured by the composite score for magnitude comparison across notations, is a unique predictor of estimation ability, often above and beyond general mathematical ability and number line estimation. In particular, students with higher integrated number sense are more than twice as likely to correctly answer the aforementioned 12/13+7/8 estimation problem than their peers with the same number line estimation ability and general math ability. This finding suggests that integrated number sense is an important inhibitor for impulsive calculation, above estimation ability for individual fractions and a general standardized test of math achievement. Finally, this dissertation advances the field of mathematics education by suggesting instruction that connects equivalent values with varied notations might provide superior benefits over a sequential approach to teaching rational numbers. At a minimum, this dissertation suggests that more careful attention must be paid to relating rational number notations. Future work might examine the origins of impulsive calculation and the observed percentages-are-larger bias. Future research might also examine whether integrated number sense is predictive of estimation ability beyond general number sense within notations. From these investigations, it might be possible to design a more impactful intervention to improve rational number outcomes.

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