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...

Evaluating and Improving a Learning Trajectory for Linear Measurement in Elementary Grades 2 and 3: A Longitudinal Study

An egg drop contest recently held at the University of Witwatersrand served to test 
engineering students' ability to solve non-standard problems. In this contest, students 
designed and built structures to protect their cargo - a raw egg. The structures were 
dropped 15 m onto a hard surface. The goal was to prevent damage to the egg upon 
impact. This paper explores some of the experimental and theoretical techniques 
that could be used to analyse and design a suitable impact resistant structure. The 
accelerations of the egg during free-fall and impact were measured by a miniature custom 
built accelerometer sensor and recorded by a data acquisition system. Measurements 
show that an egg housed in a test structure and dropped from only 3 m can produce 
accelerations greater than 70 g. The results from a theoretical model of impact presented 
in this paper closely matches the measured accelerations. The model predicts that if the 
test structure investigated would be dropped from 15 m, the egg would be subject to 
accelerations of approximately 100 g.

Developing understanding of structural implications of dynamic loading in engineering students through the egg drop contest - theory and measurements : technical paper

/pdf/children-s-conceptions-of-area-measurement-and-their-3wiojudeuh.pdf

Children’s Conceptions of Area Measurement and Their Strategies for Solving Area Measurement Problems

The Role of Theories of Learning and Cognition in Assessment Design and Development

Many studies showed that enormous students got confused about the concepts of measuring area of circles. The main reason is because mathematics classroom practices emphasized on memorizing formulas rather than understanding concepts. Therefore, in this study, a set of learning activities were designed as an innovation in learning area measurement of circles. The activities involved two models namely grid paper and reshaping which are respectively as a means and a strategy to support students' learning of area measurement of circles. Design research was used as the research approach to achieve the aim. Thirty-eight of 8th graders in Indonesia were involved in this study. In this study, together with the contextual problems, the grid paper and reshaping sectors, which used as the models in this learning, helped the students to gradually develop their understanding of the area measurement of circles. The grid papers plays important role in comparing and estimating areas. Whereas, the reshaping sectors might support students' understanding of the circumference and the area measurement of circles. Those two models could be the tool for promoting the informal theory of area measurement. Besides, the whole activities gave important role on distinguishing the area and perimeter of circles.

https://iopscience.iop.org/article/10.1088/1742-6596/948/1/012058/pdf

Models to support students’ understanding of measuring area of circles

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.

Children’s unit concepts in measurement: a teaching experiment spanning grades 2 through 5

Advancing fourth-grade students' understanding of arithmetic properties with instruction that promotes mathematical argumentation

We describe how mathematical argumentation supports curiosity and exploration by sharing a first-grade lesson in which students explored decomposition with subtraction. We also reflect on the conditions that supported the inclusion of mathematical argumentation.

Chepina Witkowski Rumsey

Papers

Children’s unit concepts in measurement: a teaching experiment spanning grades 2 through 5

Advancing fourth-grade students' understanding of arithmetic properties with instruction that promotes mathematical argumentation

Subtraction, Decomposition, and Argumentation