Size and scale are crosscutting ideas integral to scientific understanding. However, research shows that students have little understanding of the size of objects, particularly objects too small to see with the unaided eye. Using a cross-sectional study with 101 middle-school through undergraduate students, a teaching experiment with 24 middle school students, and a theoretical task analysis, I built a learning progression for one-dimensional size and scale that focused on four aspects: ordering by size, grouping by size, relative scale (how many times bigger one object is than another), and absolute size. Over 90% of students develop their conceptual understanding of size and scale by connecting the aspects of size and scale in a specific order. Learners first connect the two qualitative aspects: ordering and grouping. They next connect ordering and relative scale, and then connect ordering and absolute size. Learners connect the two quantitative aspects, relative scale and absolute size, last.

A major assumption underlying the learning progression theory is that they can be used to design learning materials that will advance student understanding. To provide empirical support for this idea, a teaching experiment was conducted to determine if students could advance in the manner suggested by the learning progression. The teaching experiment resulted in statistically and educationally significant learning gains. It showed that students increase the accuracy of their factual knowledge in tandem with their increased connectedness of knowledge, by establishing landmark objects that help them construct a mental measurement line. Measurement units including micrometers and nanometers were shown to be powerful tools for student learning about the unseen world.

The task analyses revealed a variety of strategies that learners can use in addressing size and scale tasks. Types of strategies include using recall, using a single aspect, and employing the connection across aspects. Required logical and mathematical skills that precede proportional reasoning were also identified.

This dissertation provides a model of how to develop a learning progression for a core idea, and showed that a learning progression can inform the design of curriculum materials that can move students to more advanced levels on the learning progression.