We examined children's development of strategic and conceptual knowledge for linear measurement. We conducted teaching experiments with eight students in grades 2 and 3, based on our hypothetical learning trajectory for length to check its coherence and to strengthen the domain-specific model for learning and teaching. We checked the hierarchical structure of the trajectory by generating formative instructional task loops with each student and examining the consistency between our predictions and students' ways of reasoning. We found that attending to intervals as countable units was not an adequate instructional support for progress into the Consistent Length Measurer level; rather, students must integrate spaces, hash marks, and number labels on rulers all at once. The findings have implications for teaching measure-related topics, delineating a typical developmental transition from inconsistent to consistent counting strategies for length measuring. We present the revised trajectory and recommend steps to extend and validate the trajectory. Measurement of geometric space is an important aspect of the broad curriculum of mathematics in elementary education, yet research on teaching and learning measurement is less robust and less extensive than research on other domains such as number. Further, the research on learning measurement has not been thoroughly integrated with instructional design and pedagogical concerns. The goal of this study is to implement and improve a learning trajectory in the domain of measurement. We are particularly interested in the potential integration of developmental progressions that sequence increasingly sophisticated types of thinking in a domain with pedagogical progressions that list appropriate types of tasks and instructional activities suited to specific levels along that developmental progression. By integrating learning progressions and pedagogical progressions within the context of theoretical accounts of children's cognitive operations on mental images and concepts, we expect to forge more practical and productive learning trajectories. Such trajectories provide important infrastructure for improving assessment tools, professional development models, and curriculum design.
We examined ways of improving students' unit concepts across spatial measurement situations. We report data from our teaching experiment during a six-semester longitudinal study from grade 2 through grade 5. Data include instructional task sequences designed to help children (a) integrate multiple representations of unit, (b) coordinate and group units into higher-order units, and (c) recognize the arbitrary nature of unit in comparison contexts and student's responses to tasks. Our results suggest reflection on multiplicative relations among quantities prompted a more fully-developed unit concept. This research extends prior work addressing the growth of unit concepts in the contexts of length, area, and volume by demonstrating the viability of level-specific instructional actions as a means for promoting an informal theory of measurement.
This study examines the difficulties college students experience when creating and interpreting graphs in which speed is one of the variables. Nineteen students, all preservice elementary or middle school teachers, completed an upper-level course exploring algebraic concepts. Although all of these preservice teachers had previously completed several mathematics courses, including calculus, they demonstratedwidespread misconceptions about the variable speed. This study identifies four cognitive obstacles held by the students, provides excerpts of their graphical constructions and verbal interpretations, and discusses potential causes for the confusion. In particular, misconceptions arose when students interpreted the behavior and nature of speedwithin a graphical context, as well as in situations where they were required to construct a graph involving speed as a variable. The study concludes by offering implications for the teaching and learning of speed and its interpretation within a graphical setting.
This study explored children’s area estimation performance. Two groups of fourth grade children completed area estimation tasks with rectangles ranging from 5 to 200 square units. A randomly assigned treatment group completed instructional sessions that involved a conceptual area measurement strategy along with numerical feedback. Children tended to underestimate areas of rectangles. Furthermore, rectangle size was related to performance such that estimation error and variability increased as rectangle size increased. The treatment group exhibited significantly improved area estimation performance in terms of accuracy, as well as reduced variability and instances of extreme responses. Area measurement estimation findings are related to a Hypothetical Learning Trajectory for area measurement.
We describe several service-learning initiatives implemented by the mathematics and education departments. College students with majors and minors in math and math education have helped to design and implement math events for elementary and middle school students. Formal and informal reflections on these service-related experiences have demonstrated the potential impact on future teaching and learning goals for the pre-service teachers. College students taking a statistics course have analyzed data from some of these math events, using both descriptive and inferential methods. Teachers of participating elementary and middle schools have re-examined their textbook and curriculum choices in light of their students' performance at these math events.
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