Additive and Multiplicative Reasoning
Lamon,15 whose work on proportional reasoning and rational number has made a great contribution to our understanding of students’ learning, elucidates the distinction between relative and absolute reasoning. She asks the learner to consider the growth of two fictitious snakes: String Bean, who is 4 feet long when the story begins, and Slim, who is 5 feet long. She tells us that after 5 years, both snakes have grown. String Bean has grown from 4 to 7 feet, and Slim has grown from 5 to 8 feet (see the figure below). She asks us to compare the growth of these two snakes and to answer the question, “Who grew more?”
Lamon suggests that there are two answers. First, if we consider absolute growth, both snakes grew 3 feet, so both grew the same amount. The second answer deals with relative growth; from this perspective, String Bean grew the most because he grew 3/4 of his length, while Slim grew only 3/5 of his length. If we compare the two fractions, 3/4 is greater than 3/5, and so we conclude that String Bean has grown proportionally more than Slim.
Lamon asks us to note that while the first answer, about the absolute difference, involves addition, the second answer, about the relative difference, is solved through multiplication. In this way she shows that absolute thinking is additive, while relative thinking is multiplicative.