Proportional reasoning

Proportional Reasoning

Proportionality is a mathematical relation between two quantities. Proportional reasoning is one of the skills a child acquires when progressing from the stage of concrete operations to the stage of formal operations according to Piaget’s theory of intellectual development.

What is Proportionality?

If you look up proportionality in wikipedia, one of the definitions you find is “In mathematics and in physics, proportionality is a mathematical relation between two quantities.” There are two different views of this “mathematical relation”; one is based on ratios and the other is based on functions.

An Arithmetic Viewpoint

In many school books proportionality is expressed as an equality of two ratios:

${\frac {a}{b}}={\frac {c}{d}}$

Given the values of any three of the terms, it is possible to solve for the fourth term. Once a student has mastered this arithmetic skill, one is tempted to think the student understands proportional reasoning. Sadly, experimental evidence clearly indicates this may not be the case.

A Functional Viewpoint

A scientist has a much different view of proportionality. Given the following equation for the force of gravity (according to Newton)

$F=G{\frac {m_{1}m_{2}}{r^{2}}}$

the scientist would say that the force of gravity between two masses is directly proportional to the product of the two masses and inversely proportional to the square of the distance between the two masses. From this perspective proportionality is a functional relationship between variables in a mathematical equation.

Proportional Reasoning and Intellectual Development

In Piaget’s model of intellectual development, the fourth and final stage is the formal operational stage. In the classic book “The Growth of Logical Thinking from Childhood to Adolescence” by Jean Piaget and Barbel Inhelder formal operational reasoning takes many forms, including propositional reasoning, deductive logic, separation and control of variables, combinatorial reasoning, and propositional reasoning. Robert Karplus, a world-renown science educator in the 1960s and 1970s, investigated all these forms of reasoning in adolescents and adults, but he is perhaps best know for his study of proportional reasoning.

The Mr. Tall-Mr. Short Problem of Robert Karplus

Problem Statement

Here is a picture of Mr. Tall and Mr. Short.

Image of Mr. Tall and Mr. Short.

Mr. Short is six paper clips in height. If he is measured in large buttons he is four large buttons in height.

Mr. Tall is similar to Mr. Short but is six large buttons in height.

Predict the height of Mr. Tall if you could measure him in paper clips. Explain your response.