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This work started as a discussion between Steve Kanim, Suzanne Brahmia and Andrew Boudreaux who had all been trying independently to better understand students' reasoning about proportions and who were developing curricular materials to help their students develop better mathematical reasoning skills. We believe that what is typically thought of as student learning in physics is actually just the tip an iceberg with a much larger base of mathematical sensemaking that is largely unseen and under-assessed as we invite our students to learn the practices of physics. As we began our struggle to articulate what entails mathematical sensemaking in physics, we recognized that the project was bigger than three researchers with curious minds could manage unaided. We subsequently received funding from the NSF, and the work took on a life of its own. In addition to the researchers whose work we cite below, our work is strongly influenced by the philosophical underpinnings of Arnold Arons and Robert Karplus.

Our invention tasks are based on an instructional method developed by Dan Schwartz and colleagues at Stanford. Our initial use of invention instruction challenges students to identify the key features of a situation and invent a method of numerical characterization that allows meaningful comparisons to be made. During this process, students examine contrasting cases, situations that are recognizably similar, yet differ in important ways. For example, students might be shown three different archery targets, each with five arrows embedded at various places, and asked to come up with a way to characterize and compare the skill of the archers. Invention tasks are open-ended and tap into students' natural sense-making abilities. Recognizing a need and formulating a response leads students to build their own knowledge that is both durable and usable. Invention instruction has been shown to prepare students for subsequent direct instruction.

We develop tasks in which students use data from contrasting cases to invent ratio or product quantities, rules or equations to characterize a variety of physical systems. Students work through sequences of such tasks to ramp up from everyday contexts to more abstract physics contexts. For example, students might be given data about various popcorn poppers, and asked to invent a number that characterizes how fast the popper fills a bowl with popcorn. Working in groups, students invent a ratio that describes the number of kernels that are popped each second. Next, the students consider the motion of cars on a road, and use analogous reasoning to invent a "fastness index." In this sequence, students generate the concept of time rate of change in an everyday context, then work with its ratio representation before encountering formal instruction on velocity. We have field tested sets of invention tasks, called invention sequences, both at the pre-college level, in middle school and high school, and in a variety of introductory physics courses, from pre-service teachers to engineering students.