Evaluating Quantitative Reasoning Strategies for Comparing Fractions: A Tool for Teachers

Abstract

Meaningful inclusion of quantitative reasoning into mathematics instruction requires meaningful ways to evaluate it. Few formative assessments exist to evaluate the strategies students use when reasoning mathematically. The Framework for Evaluating Quantitative Reasoning Strategies presented in this article provides teachers with categories for evaluating types of quantitative reasoning strategies students use for problems in the mathematical domain of number and operations. Numerous examples of the types of strategies students use for comparing fractions and how to evaluate the complexity of these strategies are provided. Included are research-based instructional recommendations for moving students toward a deeper conceptual understanding of fractions as numbers.

Keywords

intervention academic assessment instruction mathematics

Two fifth-grade students with mathematics learning disabilities (MLD) were asked to identify which of two fractions, $\frac{3}{8}$ and $\frac{5}{6}$ , was larger and be prepared to justify their answer. One student explained, “Because 8 has smaller pieces and 6 is big chunks, and 3 smaller pieces is not as big as 5 big chunks, I think $\frac{5}{6}$ is larger than $\frac{3}{8}$ .” A second student responded, “My teacher taught me a strategy called the parachute, first you have to times the denominator to the other fraction’s numerator, so 6 × 3 and then 8 × 5, and then 8 × 5 is 40 and 3 × 6 is 18 and 40 is bigger than 18, so $\frac{5}{6}$ is bigger.” Both students arrived at the correct answer, but their explanations demonstrated very different quantitative reasoning strategies. A framework is presented in this article for categorizing types of quantitative reasoning strategies, and a discussion is provided about how this framework can be used, in the area of fractions, to inform instruction for students with disabilities or difficulties in mathematics.

Instruction in the field of mathematics for low-performing students often focuses on basic skills and simple computation (van Garderen, Scheuermann, Jackson, & Hampton, 2009). Similarly, strategy instruction, found to be effective with improving the ability of many low-performing students to perform mathematical procedures (Montague, 2003), tends to rely on a formulaic approach to working with numbers and operations and does not always prompt students to reason mathematically (i.e., think about the numbers and operations). Quantitative reasoning, on the other hand, requires students to pay attention “to the meaning of quantities, not just how to compute them” (National Governors Association Center for Best Practices & Council of Chief State School Officers [CCSSO], 2010, p. 6). The Common Core State Standards for Mathematical Practice stress the equal value of both procedural knowledge and conceptual understanding (CCSSO, 2010) as well as emphasize the importance of mathematical reasoning at every grade level. All students, including students with MLD, should be provided with opportunities to reason mathematically and to communicate their reasoning both verbally and in writing (Gersten et al., 2009; National Council of Teachers of Mathematics [NCTM], 2014). Teaching basic skills to students, in the absence of opportunities to reason mathematically, will fail to prepare them for the demands of high school or college mathematics (Tate & Johnson, 1999). Therefore, teaching mathematics to students with MLD or learning difficulties in mathematics must include instruction on how to reasoning quantitatively.

Meaningful inclusion of quantitative reasoning into mathematics instruction requires meaningful ways to evaluate it (CCSSO, 2010). Yet, unlike formative assessments of basic computation skills, such as AIMSweb (Pearson, 2014) or easyCBM (Houghton Mifflin Harcourt, n.d.), there are few formative assessments that evaluate those strategies students use when reasoning mathematically. One of these tools is the Math Reasoning Inventory (MRI; Burns, 2012) and is available at www.mathreasoninginventory.com. Unfortunately, the MRI provides 36 different categories for types of student reasoning, depending upon the specific problem posed. Unlike the MRI, the Framework for Evaluating Quantitative Reasoning Strategies (Crawford, Huscroft-D’Angelo, Quebec Fuentes, & Higgins, 2018) provides a total of five categories for evaluating types of quantitative reasoning strategies students use for problems in the mathematical domain of number and operations. In particular, this article provides an example of how teachers can use this framework to evaluate strategies students use when comparing fractions, as understanding fraction equivalence and how to order fractions from least to greatest are important precursors for understanding operations with fractions, an often-cited challenge for students with MLD (Mazzocco, Myers, Lewis, Hanich, & Murphy, 2013; Tian & Siegler, 2017).

Reasoning about Fractions

Fractions are an integral part of the school mathematics curriculum (Barnett-Clarke, Fisher, Marks, & Ross, 2010; Siegler et al., 2010; Smith, 2002), and evidence has shown the importance of fraction understanding for later success in mathematics (Hecht & Vagi, 2010; Mazzocco et al., 2013; Ross & Bruce, 2009; Siegler et al., 2012). Students struggle, however, with understanding the meaning of fractions (Siegler et al., 2010; Siegler & Pyke, 2013). Concepts such as the size of fractions and fraction equivalence pose challenges for students (Cramer & Whitney, 2010; Small, 2014; Smith, 2002). Misunderstandings often arise because students do not consider fractions as single numbers, but instead view them as two whole numbers (Ni & Zhou, 2005; Siegler & Pyke, 2013; Small, 2014, Smith, 2002). For example, based on the number in the denominators, students believe that $\frac{1}{3}$ is less than $\frac{1}{4}$ , which is less than $\frac{1}{5}$ (Cramer & Henry, 2002). Although fractions pose challenges for many students, they are particularly challenging for those with MLD (Mazzocco & Devlin, 2008). Students with MLD demonstrate weak fraction computation skills compared to their typical peers (Calhoon, Emerson, Flores, & Houchins, 2007), and middle school students with MLD demonstrate fewer strategies for working with fractions and less facility with generalizing fraction knowledge to more complicated problems than typically achieving and even low-achieving peers (Mazzocco et al., 2013). Mazzocco et al. (2013) suggested that students with MLD appear to use “rote performance as a backup strategy in the absence of fraction knowledge” (p. 385), demonstrating the importance of conceptual understanding related to the meaning of fractions.

Development of the Framework

The Framework for Evaluating Quantitative Reasoning Strategies (Figure 1), was validated in the mathematical domain of number and operations (Crawford et al., 2018), but because of its general categories, may also be applicable across other mathematical domains. The framework was developed in two stages. In the first study, an analysis of 1,193 responses offered by 105 low-performing students (Grades 3–5), resulted in three categories of reasoning strategies. In the second study, an additional 4,000 responses provided by a broad sample of fifth-grade students (N = 418) were analyzed, resulting in a five-category framework.

Figure 1.

Framework for Evaluating Quantitative Reasoning Strategies. Responses to the question, “Which is greater, $\frac{3}{4}$ or $\frac{4}{10}$ ?”

All of the questions for both analyses were taken from the MRI (Burns, 2012) and included whole-number computation items and fraction comparison problems. Specifically, students were asked to answer questions without the use of paper, pencil, or calculator. Students’ verbatim answers were entered into the MRI website, and each answer was followed by the question, “How did you figure that out?” These efforts resulted in development of an empirically supported and theoretically grounded (Bergqvist, 2005, 2006; Lithner, 2008) framework for evaluating students’ quantitative reasoning strategies. In the framework, the following five categories captured the full range of the nearly 6,000 student responses collected in the initial validation efforts: (a) no response/reasoning strategy, (b) unfounded reasoning strategy, (c) partial reasoning strategy, (d) algorithmic reasoning strategy, and (e) conceptual reasoning strategy (see Figure 1).

Using the Framework with Students

The Framework for Evaluating Quantitative Reasoning Strategies provides descriptions and examples for five types of quantitative reasoning strategies most employed by students in the aforementioned studies. In the first column of Figure 1, three strategy types are labeled as independent, while conceptual reasoning strategies and procedural reasoning strategies are labeled as reciprocal because conceptual understanding and procedural fluency are two of five integrated components of proficiency in mathematics (Kilpatrick, Swafford, & Findell, 2001). The NCTM (2014) argues that procedural fluency develops from a strong conceptual understanding. At times, however, students can rely solely on deep and nuanced procedural strategies (Nunes et al., 2015; Star, 2005) to reason, while for other problems (depending on content and complexity), procedural knowledge and conceptual knowledge may act bidirectionally (Hecht & Vagi, 2010; Siegler & Stern, 1998). See Figure 1 for a definition of each type of strategy, along with an example response for each strategy type. A more thorough discussion of all five categories is provided next, along with additional examples of reasoning that students demonstrate when comparing two fractions, or ordering three fractions, from least to greatest. These additional examples were collected during a study that allowed the use of traditional and electronic writing tools.

No Response/Reasoning Strategy Provided

The definition of this category is straightforward—no reasoning strategy is apparent in the response, or lack thereof, shared by the student. A large majority of the responses coded into this category are “I don’t know.” Some argue that no incorporation of a reasoning strategy may in fact be better than the use of an unfounded reasoning strategy which could be a reflection of ingrained mathematical misconceptions that can be hard to remediate (Barbash, 2012). On the other hand, errors, or misconceptions, are often viewed positively as they provide an opening for investigation and learning (Ashlock, 2010). Important for use of this framework is that the instructional decisions made by teachers will vary based on whether or not a student attempts to apply a strategy or not.

Unfounded Reasoning Strategy

An unfounded reasoning strategy is one that is based on a logical fallacy or application of an inappropriate strategy. Figure 2 provides two different illustrations of an unfounded reasoning strategy. The first example in Figure 2 demonstrates whole-number thinking that does not attend to the different sizes of the “4 pieces” (e.g., ninths and fifths) and results in an incorrect answer. As mentioned in the introduction, whole-number reasoning is a strategy used by many students. Teachers must provide targeted instruction to help students understand that a simple fraction represents one part of a whole and not two separate whole numbers. The second response in Figure 2 is incorrect and based on an erroneous application of the common denominator algorithm. When discussing the unfounded reasoning strategy category, the correctness of an answer is not the focus. This decision was made for three reasons. First, mathematics educators discuss mathematical reasoning as something that evolves over time through the sharing of one’s thoughts with others, regardless of whether or not this reasoning concludes in a correct answer (Russell, 1999). Second, the problems could be answered correctly due to guessing (i.e., part of the definition of unfounded). Furthermore, an answer can be correct, but still demonstrate inaccurate conceptual understanding; this combination will prove to be an ineffective strategy as the student progresses toward more abstract mathematics.

Figure 2.

Examples of responses coded as unfounded reasoning strategies.

Partial Reasoning Strategy

A partial reasoning strategy reflects reasoning that makes sense but is underdeveloped. In other words, the response is “heading in the right direction,” but is just not fully developed. In this way, it differs from use of an unfounded reasoning strategy (i.e., either partially or fully developed). Figure 3 provides three student responses that were coded as using a partial reasoning strategy. Importantly, in the first example, the student’s answer would be correct if the two fractions had the same numerator, but because they did not and the student did not extend her reasoning as demonstrated in the opening paragraph, her reasoning is categorized as partial. In the next response (Example 2), the student also did not make reference to the numerators being equal as well as the wholes being the same size. This response is coded as “partial” due to the accuracy of its picture but the absence of an explanation for the difference in the size of the shaded regions. Finally, the last response was categorized as partial because the student’s reasoning is moving in the right direction as she attempts to use the number of pieces in conjunction with the size of the pieces to compare the fractions.

Figure 3.

Examples of responses coded as partial reasoning strategies.

Algorithmic Reasoning Strategy

According to Bergqvist (2005, 2006), algorithmic reasoning is based exclusively on applying a procedure or a memorized response. Although application of this procedure may often lead to a correct answer, one cannot assume that a student understands the number and operation concepts behind the answer. Students who employ algorithmic reasoning strategies not only solve the problem using a set of specific procedures but also defend their answers through reiteration of the procedure used. Figure 4 contains examples of student responses, to comparing fractions, coded as algorithmic reasoning strategies. In both instances, the responses represent the procedure of cross-multiplying and comparing the products to compare the fractions. The procedure is used in the absence of shared reasoning about why it works and reinforces whole-number thinking about fractions.

Figure 4.

Examples of responses coded as algorithmic reasoning strategies.

Conceptual Reasoning Strategy

Use of a conceptual reasoning strategy implies an inherent understanding of numbers and their relationships. Figure 5 contains examples of conceptual reasoning strategies. Both students compare the given fractions to the $\frac{1}{2}$ benchmark and use the transitive property (i.e., since $\frac{3}{4}$ is greater than $\frac{1}{2}$ and $\frac{4}{10}$ is less than $\frac{1}{2}$ , $\frac{4}{10}$ is less than $\frac{3}{4}$ ) to arrive at a conclusion. The first student also demonstrates flexibility with representation of rational numbers by incorporating percentages.

Figure 5.

Examples of responses coded as conceptual reasoning strategies.

Use of a conceptual reasoning strategy may or may not include algorithmic reasoning strategies. Depending on the context and the complexity of the problem posed, sometimes both procedural and conceptual knowledge are drawn on to best solve a problem. Because of the often-cited bidirectional relationship between procedural and conceptual knowledge (Bailey et al., 2015; CCSSO, 2010), teachers need to differentiate between responses that reflect a procedure and can be further elaborated upon using number and operation sense and those responses that reflect a procedure defended only by a reiteration of the procedure used.

Instructional Guidance in Response to Reasoning Strategies

Finally, the strong recommendations implied throughout this article are that students make their thought processes public (NCTM, 2014), allowing teachers to build upon (scaffold) this thinking to promote deeper understanding (Lenz, 2006; McLeskey et al., 2017). Figure 6 provides suggestions for how to scaffold student thinking based on the initial categorization of students’ reasoning strategies as well as specific teacher questions in the context of the examples presented throughout the present article. The questions serve as a means to start conversations by encouraging students to share their thinking, fostering evaluation of reasoning strategies, and inquiring about and furthering student thinking (NCTM, 2014).

Figure 6.

Possible teacher responses to each type of reasoning strategy.

As with other students, those with MLD or with difficulties in mathematics, must be provided opportunities to verbalize their reasoning (Gersten et al., 2009) when solving mathematical problems (including comparison of fractions). Approaches, such as small group instruction, can facilitate mathematical discourse (NCTM, 2014), about quantitative reasoning as students share their thinking using either words, pictures, and/or text. Peer discussion about the use of quantitative reasoning strategies can be a powerful tool (Goos, Galbraith, & Renshaw, 2002; Kunsch, Jitendra, & Sood, 2007; NCTM, 2014). Figure 7 includes other research-based instructional considerations to help students build foundational understanding of fractions as numbers.

Figure 7.

Instructional considerations when teaching about fractions.

The end goal for teachers is to move students’ use of the no response/reasoning strategy or an unfounded reasoning strategy to the application of a strategy based on an evolving sense of number and operations, even if it is not fully developed and does not always result in a correct answer. to enhance students’ reasoning skills, one must first evaluate their understanding of the problem types at hand, and then use research-based methods (such as the ones illustrated in Figure 7), for helping students quantitatively reason about numbers. Students who have been taught the use of an acronym to mechanically work through a problem involving fractions to arrive at a correct answer, with limited conceptual understanding, will have little success with more complex mathematics.

Conclusion

Part of becoming proficient in mathematics involves the ability to reason, communicate mathematical thinking, and engage in the process of constructing new knowledge (NCTM, 2000). However, evaluating student reasoning and making connections to instructional practices is challenging. The ongoing emphasis on mathematical reasoning both in research and practice raises the need for the development of the framework presented here.

Considering the two examples of quantitative reasoning shared in the first paragraph, how might one categorize those responses according to the five-categories in the Framework for Evaluating Quantitative Reasoning Strategies, and how might this inform instruction? Using this framework, teachers can decipher how students think about quantitative problems, in the domain of Number and Operations. For example, teachers may design different types of instruction for students who rely on partial reasoning strategies about the relative size of two fractions and those students who provide responses that gravitate toward algorithmic reasoning strategies in the absence of clear conceptual understanding.

The incorporation of mathematical reasoning across many of the Common Core State Standards has resulted in a renewed instructional focus on reasoning. Yet, as NCTM (2014) states, “Standards do not teach; teachers teach” (p. 1). Hand in hand with teaching goes formative evaluation, and teachers may benefit from using the Framework for Evaluating Quantitative Reasoning Strategies as they collect data on the progress of their students’ quantitative mathematical reasoning. This information can then be used to design instruction and/or implement interventions to change the trajectory of student outcomes in the area of fractions.

Footnotes

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This project was funded through the Mathematics e Text Research Center (MeTRC), at the University of Oregon.

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