Research Foundations

Existing research informs the development of EM2 assessments

The importance of understanding rational number concepts

The EM2 Project was focused on grades 3-5 rational number conceptions because rational number understanding is a critical foundation for success in algebra, which is a prerequisite for access to and success in higher-level mathematics. Therefore, having rational number misconceptions could have serious consequences for students' mathematics course-taking in middle school and high school, as well as their college and career readiness

Many students struggle with rational number concepts

Researchers and practitioners report that students and adults in the United States have considerable difficulty with fractions, decimals, and percents. When middle school and high school teachers are asked what content students struggle with most, they often report rational number concepts. Student difficulty with rational number concepts is repeatedly seen on results of the National Assessment of Educational Progress (NAEP). In 2007 only 46% of 8th graders across the nation could add three decimal fractions (7/10, 7/100, 7/1000) and represent the sum as a decimal, and only 49% of 8th graders nationally accurately ordered the following fractions from least to greatest: 2/7, 1/2, 5/9. In 2007, Susan Lamon wrote, “Of all the topics in the school curriculum, fractions, ratios, and proportions arguably hold the distinction of being the most protracted in terms of development, the most difficult to teach, the most mathematically complex, the most cognitively challenging, the most essential to success in higher mathematics and science, and one of the most compelling research sites” (The Second Handbook of Research on Mathematics Teaching and Learning: A Project of the National Council of Teachers of Mathematics, p. 629).


Rational number concepts are very complex, and if not mastered can create major barriers for students’ mathematical development. Many difficulties and errors with rational number concepts derive from common conceptual flaws that stem from the inherent properties of rational numbers and the transition from learning about and working with the whole number system. While whole number relationships are based on additive properties, rational numbers have relationships based on multiplicative relations. Moreover, rational numbers can be expressed in many different forms and can be designated by an infinite number of equivalent representations. Two numbers that students understood as functioning independently can now compose a new number--a common fraction--that has a distinct value. To add further complication, a single rational number can be representative of several distinct conceptual meanings. These distinct conceptual meanings are referred to as “sub-constructs” of rational number such as part-whole relation (4 of 5 equal shares), quotient interpretation (implied division, 2 submarine sandwiches divided by 3 boys), measure (fixed quantity on a number line), ratio (5 girls to 6 boys), and multiplicative operator (scaling: reduce or enlarge).

Student misconceptions related to rational numbers

The EM2 Project drew on currently available research that has found a significant number of middle-level student misconceptions and conceptual issues specific to the three EM2 Project areas: fractions, decimals, and operations with fractions and decimals.


One commonly held misconception is when students do not understand the meaning of the fraction symbol, and instead focus on either the numerators or denominators when ordering or comparing common fractions. When comparing two fractions such as 1/2 to 1/4, they will compare 2 and 4. Students with this conceptual flaw may conclude that since 4 is greater than 2, then 1/4 is greater than 1/2. The EM2 Comparing Two Fractions assessment is designed to diagnose whether students have this misconception, as well as other misconceptions related to comparing two fractions.

Other examples of common misconceptions are related to how students think about adding rational numbers. They may add the two fractions 1/4 and 1/3 and think the solution is 2/7 because they first add the numerators together and then add the denominators together. Similarly, in the case of decimals, they may add 1.5 and 0.23 to get a sum of 38.


Students may also have difficulty with fact that the two numbers composing a common fraction--the numerator and denominator--are related through multiplication and division, not addition. For example, when when comparing fractions students may focus on the difference between the numerator and denominator to come to an inaccurate conclusion. When looking at 3/4 and 5/6, they see that in each case the difference between numerator and denominator is one and incorrectly conclude that these two fractions are equal.

The need to identify student misconceptions

Students come to the classroom with preconceived ideas about mathematics topics. Research on cognitive development tells that as students learn new information, they must integrate it into previously held knowledge. In other words, learning is not simply acquiring new facts, but involves incorporating new knowledge into existing knowledge. Recommendations from an important report from the National Research Council, How People Learn (Bransford, Brown, & Cocking, 2000), emphasize the need to build on existing knowledge and to engage students’ preconceptions, particularly when those preconceptions interfere with learning. This is because when new new information conflicts with how students' current understanding, the preconceptions can serve as a barrier to constructing new knowledge.


Therefore, to support student learning, teachers must identify existing student conceptions and adjust instructional activities to directly and explicitly confront elicited misconceptions. Research evidence tells us that diagnosing a student's specific area of difficulty can lead to more effective instruction, especially when coupled with specific intervention tools and strategies. If teachers do not have a clear understanding about how students are thinking about a particular concept, they may not present new information in ways that enable students to effectively integrate it into their existing thinking and the result is likely to be that the intended learning will not occur.

The role of EM2 assessments in teachers formative assessment practice

Research highlights the usefulness of teachers' formative assessment practice, which varies greatly from traditional summative and large-scale assessment. Formative assessment includes “all those activities undertaken by teachers, and/or by their students which provide information to be used as feedback to modify the teaching and learning activities in which they are engaged” (Black & Wiliam, 1998, p. 10). Researchers and practitioners have investigated the topic more deeply, expanded the definition, and clarified the attributes of formative assessment to further define it as the systematic iterative process of gathering and analyzing evidence, providing feedback to students, altering instruction, and gathering more evidence to map from the current conceptions to a targeted learning goal. When formative assessment is integrated into daily classroom practice, targets critical goals, and is on-going, it can become a powerful tool to enhance learning.


The EM2 diagnostic assessments help teachers identify which of their students are likely to have specific types of rational number misconceptions. Teachers can then use this information to inform their instruction.

Prior work informs the use of assessments to elicit misconceptions

The EM2 Project built on Pam Buffington's and Cheryl Tobey's extensive work with teachers. Previous projects include the Northern New England Co-Mentoring Network and Curriculum Topic Study, both of which were funded by the National Science Foundation. Tobey and Buffington also lead two Maine State Math Science Partnerships: Reducing Barriers by Addressing Mathematics Misconceptions and Addressing Access to Algebra using Interactive Technologies. In addition, Tobey and her colleagues have published a number of books on formative assessment probes, including Uncovering Student Thinking About Mathematics in the Common Core, Grades 6-8 (Tobey & Arline, 2013).