Learning to Quantify Relationships Among Weight, Size, and Kind of Material

ABSTRACT

In a 1-hour teaching interview, 20 children (aged 7 to 11) discovered how to tell whether objects might be made of the same material by using ratios of measures of weight and size. We examine progress in the children's reasoning about measurement and proportional relations, as well as design features of instruments, materials, and tasks crafted to enable such progress. Findings are discussed in light of theories and research about cognitive development and learning.

Acknowledgments

This study was partially supported by the National Science Foundation through grants #0628245 (The Inquiry Project) and #0962863 (The Poincaré Institute for Mathematics Education). The research originated from the first author's PhD Dissertation in Science and Mathematics Education at Tufts University (Liu, 2012). We thank Roger Tobin, Bárbara Brizuela, Jane Greenspan, and Tracy Noble for suggestions and feedback at various stages of the study. We also thank all the children, who so thoughtfully considered our interview questions.

Notes

1. We use quantity throughout to refer to a “property of a phenomenon, body, or substance, where the property has a magnitude that can be expressed as a number and a reference” (BIPM et al., 2012). The reference is typically a unit of measure or a unit for counting uniform elements in a collection (e.g., number of oranges).

2. As we have noted elsewhere (Carraher & Schliemann, 2007), both approaches entail the same function, albeit through different expressions. The ratio-of-unlike-quantities approach easily gives rise to closed-form expressions whereas the ratio-of-like-quantities approach is more consistent with iterative or recursive expressions.

3. This process is referred to as a metric (Weisstein, 2016) or distance function. A metric is a function that associates numbers to the difference of any two pairs of elements in a set. When values are represented as points on a number line or axis, the distance between two points corresponds to the difference in the two respective magnitudes of the quantity.