“…Based on the kappa ( K ) statistic, overall rater agreement (inter-rater reliability) was either perfect or almost perfect (Landis & Koch, 1977) for the five bar tasks: B1( K = .95); B2 ( K = 1.00); B3 ( K = 1.00); B4 ( K = .98); B5 ( K = .93), representing high reliability for rater agreement. These tasks are not dissimilar to those used by other researchers to study unit coordination (Kosko & Singh, in press; Norton, Boyce, Phillips, Anwyll, Ulrich, & Wilkins, 2015; Norton, Boyce, Ulrich, & Phillips, 2015), but they are different enough that we did not feel comfortable using them to categorize students according to the stages of unit construction and coordination. Therefore, we instead developed some hypotheses regarding their difficulty relative to each other and relative to the other tasks that we can now test.Fig.…”
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.
“…Based on the kappa ( K ) statistic, overall rater agreement (inter-rater reliability) was either perfect or almost perfect (Landis & Koch, 1977) for the five bar tasks: B1( K = .95); B2 ( K = 1.00); B3 ( K = 1.00); B4 ( K = .98); B5 ( K = .93), representing high reliability for rater agreement. These tasks are not dissimilar to those used by other researchers to study unit coordination (Kosko & Singh, in press; Norton, Boyce, Phillips, Anwyll, Ulrich, & Wilkins, 2015; Norton, Boyce, Ulrich, & Phillips, 2015), but they are different enough that we did not feel comfortable using them to categorize students according to the stages of unit construction and coordination. Therefore, we instead developed some hypotheses regarding their difficulty relative to each other and relative to the other tasks that we can now test.Fig.…”
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.
“…Engagement in multiplicative understandings of fraction quantities may be promoted by supporting and extending students’ multiplicative coordination of units. Units coordination refers to how students create units and maintain relationships with other units (Boyce & Norton, 2016; Norton et al, 2015). Norton et al (2015) explain that a student uses one level of units when he conceives of situations such as five iterations of four by counting on from the first or second set by ones and double-counting the number of fours to reach a stop value (e.g., 4, 8, 12, 13–14–15–16, then 17–18–19–20).…”
Section: Tasks To Support Productive Engagement In Multiplicative Fra...mentioning
Productive engagement in fractional reasoning is essential for abstracting fundamental algebraic concepts vital to college and career success. Yet, data suggest students with learning disabilities (LDs), in particular, display pervasive shortfalls in learning and mastering fraction content. We argue that shortfalls in understanding are in fact issues of access in terms of opportunities that students have to productively engage with learning objects (i.e., tasks) that meaningfully bring forward and promote students’ fractions understanding. In this study, we define engagement as a state and take up a single case study methodology to illustrate behavioral, affective, and cognitive engagement of Bob, a student with a LD, as he works with a series of fraction tasks designed to support his engagement. Results reveal patterns of productive engagement as regards this student’s fractional reasoning as they relate to the tasks he was given over time. Contributions of this work include insights into Bob’s engagement within tasks and provide considerations for teaching practice seeking to promote productive engagement by design.
“…The assessment on multiplicative reasoning was developed by Kosko and Singh (2018), and is based on the notion that more abstracted unit coordination is representative of more sophisticated multiplicative reasoning (Hackenberg, 2010;Norton, Boyce, Ulrich, & Phillips, 2015). The 12-item assessment used length models to assess different aspects of unit coordination (see Figure 1 for examples with included student work).…”
Section: Packet 1: Early Algebra Assessmentsmentioning
Mathematical argumentation and proof has long been identified with algebraization. Much literature discusses the relationship between the two, but with little specificity on how particular semiotic features in argumentation relate to coordination in early algebra. Further, there is a particular lack of research on this topic in the elementary/primary years of schooling. The present study examines how children's unit coordination in early algebra (particularly the concepts of equivalence and multiplicative reasoning) co-occurs with their coordination of grammatical information units. Coordination of information units was examined through reference use via the semiotic tool of detailing. Results suggest that second and third grade students who coordinate reference chains to support a mathematical claim in their argumentative writing tend to have higher multiplicative reasoning and conception of equivalence scores on several tasks. However, features of certain tasks may influence whether and how such unit coordination interacts with reference use.
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