# How Students Think When Doing Algebra, Rhine S., Harrington R., Starr C., 2019

How Students Think When Doing Algebra, Rhine S., Harrington R., Starr C., 2019.

This book is the result of a multi-year project to synthesize research on algebraic thinking from the past 35 years with primary funding from the Fund for the Improvement of Post-Secondary Education (FIPSE).

UNDERSTANDING SPEED AS RATE OF CHANGE.
Speed is commonly used to introduce students to the idea of rates of change. It is the quantification of motion—“the proportional correspondence between distance and time.”151 Students have experience with speed that they can bring to the context of a problem to help them interpret situations,152 but that experience does not always help them understand the concepts.153 It is sometimes difficult for students to understand that speed is a measure of the rate of change of two varying quantities: time and distance. Students’ experience and thinking about speed is a resource that can be improved upon.

At the beginning of their understanding of speed, students often conceive of speed as distance.154 Prospective teachers have been found to have the same confusion. 155 For example, a speed of 60 feet per second is understood as a chunk of 60 feet that can be traveled in 1 second. Although this is not incorrect, it only allows students to reason in one direction: given a speed and an amount of time, the student can add up “speed-lengths” or “rate-lengths” to find the distance. However, in order to find the amount of time it would take to cover a given distance, the only method available for students with that mindset would be to break up the distance into speed-lengths and count the number of speed-lengths in that distance.

Contents.
1. Introduction.
2. Variables and Expressions.
3. Algebraic Relations.
4. Analysis of Change (Graphing).
5. Patterns & Functions.
6. Modeling and Word Problems.

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