A Written Instrument for Assessing Students’ Units Coordination Structures

Units coordination refers to students’ abilities to create units and maintain their relationships with other units that they contain or constitute. In recent research, units coordination has arisen as a key construct that mediates opportunities for student learning across several domains of mathematics, including fractions knowledge and algebraic reasoning. To date, assessments of students’ stages of units coordinating ability have relied upon clinical interviews or teaching experiments whose time-intensive nature precludes opportunities for conducting large-scale studies. We introduce a written instrument that teachers and researchers can use with large populations of students. We report on the reliability and validity of assessments based on the instrument

BMQ

BMQ: Boston Medical Quarterly was published from 1950-1966 by the Boston University School of Medicine and the Massachusetts Memorial Hospitals

Using written work to investigate stages in sixth-grade students’ construction and coordination of units

Abstract Background Students’ ability to construct and coordinate units has been found to have far-reaching implications for their ability to develop sophisticated understandings of key middle-grade mathematical topics such as fractions, ratios, proportions, and algebra, topics that form the base of understanding for most STEM-related fields. Most of the related research on unit coordination relies on time-intensive clinical interviews and teaching experiments. In this study, we investigate the work of 93 sixth-grade students on a written assessment containing whole number and fraction contexts using both continuous and discrete quantities, and how this work can be used to assess stages in students’ construction and coordination of units. Our investigation is guided by the following general research questions: (1) What forms of written work evidence the construction of and operation on composite units (units made up of other units)? (2) How does the categorization of students based on responses from a written assessment compare to written performance on a set of tasks conveying a continuous whole number multiplicative context? Results We documented the different ways students represented composite units in their written work. In particular, student written work on tasks that included figurative unit items provided the greatest variety of evidence regarding students’ construction of and operation on composite units. However, written evidence from partitioning tasks did not seem as promising for distinguishing student stages. Students’ performance on decontextualized bar tasks involving continuous quantities was found to be consistent with students’ level of unit coordination based on written work providing evidence for the validity of stage categorizations. Conclusions Our findings shed light on the affordances and constraints associated with particular stages in unit construction and coordination that a student brings to bear on tasks provided in a formal, written assessment. These findings provide promising evidence for scaling up the assessment of students construction and coordination of units through the use of written assessments instead of time-intensive clinical interviews

Learning progression toward a measurement concept of fractions

Abstract Background Fractions continue to pose a critical challenge for students and their teachers alike. Mathematics education research indicates that the challenge with fractions may stem from the limitations of part-whole concepts of fractions, which is the central focus of fractions curriculum and instruction in the USA. Students’ development of more sophisticated concepts of fractions, beyond the part-whole concept, lays the groundwork for the later study of important mathematical topics, such as algebra, ratios, and proportions, which are foundational understandings for most STEM-related fields. In particular, the Common Core State Standards for Mathematics call for students to develop measurement concepts of fractions. In order to support such concepts, it is important to understand the underlying mental actions that undergird them so that teachers can design appropriate instructional opportunities. In this study, we propose a learning progression for the measurement concept of fractions—one that focuses on students’ mental actions and informs instructional design. Results A hierarchy of fraction schemes is charted outlining a progression from part-whole concepts to measurement concepts of fractions: (a) part-whole scheme (PWS), (b) measurement scheme for unit fractions (MSUF), (c) measurement scheme for proper fractions (MSPF), and (d) generalized measurement scheme for fractions (GMSF). These schemes describe concepts with explicit attention to the mental actions that undergird them. A synthesis of previous studies provides empirical evidence to support this learning progression. Conclusions Evidence from the synthesis of a series of research studies suggests that children’s measurement concept of fractions develops through several distinct developmental stages characterized by the construction of distinct schemes. The mental actions associated with these schemes provide a guide for teachers to design instructional opportunities for children to advance their construction of a measurement concept of fractions. Specifically, the collection of quantitative studies suggest that students need opportunities to engage in activities that support two kinds of coordinations—the coordination of partitioning and iterating, and the coordination of three levels of units inherent in fractions. Instructional implications are discussed with example tasks and activities designed to provoke these coordinations

Communicating Quantitative Literacy: An Examination of Open-Ended Assessment Items in TIMSS, NALS, IALS, and PISA

Quantitative Literacy (QL) has been described as the skill set an individual uses when interacting with the world in a quantitative manner. A necessary component of this interaction is communication. To this end, assessments of QL have included open-ended items as a means of including communicative aspects of QL. The present study sought to examine whether such open-ended items typically measured aspects of quantitative communication, as compared to mathematical communication, or mathematical skills. We focused on public-released items and rubrics from four of the most widely referenced assessments: the Third International Mathematics and Science Study (TIMSS-95): the National Adult Literacy Survey (NALS; now the National Assessment of Adult Literacy, NAAL) in 1985 and 1992, the International Adult Literacy Skills (IALS) beginning in 1994; and the Program for International Student Assessment (PISA) beginning in 2000. We found that open-ended item rubrics in these QL assessments showed a strong tendency to assess answer-only responses. Therefore, while some open-ended items may have required certain levels of quantitative reasoning to find a solution, it is the solution rather than the reasoning that was often assessed