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.

Typical Solutions

Multiplicative Reasoning 1: “He is nine paper clips tall. Each button is equal to one and a half paper clips. If he is six buttons tall you multiply six time one and a half to get nine paper clips.”

Multiplicative Reasoning 2: “Mr. Tall is 1 ½ times as high as Mr. Short. Since Mr. Short is 6 clips high, Mr. Tall must be 6 * 1 ½ = 9 clips high.”

Multiplicative Reasoning using addition: “For every two buttons there are three paper clips. Mr. Tall is 2 buttons taller than Mr. Short so he must be 3 paper clips taller. 6 + 3 = 9 paper clips.”

Additive Reasoning 1: “Mr. Tall is 8 paper clips high. Mr. Short is 4 large buttons high and 6 paper clips high. So the buttons are 2 less than the paper clips. Since Mr. Tall and Mr. Short are similar, and Mr. Tall is 6 buttons high, he must be 8 paper clips high.”

Additive Reasoning 2: “Mr. Tall is two more buttons taller than Mr. Short so he will also be two more paper clips taller than Mr. Short resulting in 8 paper clips.”

Estimate: “Nine, I figured he would be a bit taller.”

Haphazard: “Since Mr. Tall is 2 more buttons than Mr. Short, I took the 6 paper clips and multiplied by 2 to get 12 paper clips.”

For the adolescent or adult who has not attained formal operational reasoning yet, the additive solution is by far the most common. It is a consistent, logical strategy, albeit incorrect, and for the values involved in the problem it produces a believable answer. This does not appear to be a problem with arithmetic skills; in a different context these individuals can often find the value for x in the equation

${\frac {4}{6}}={\frac {6}{x}}$

Inverse Proportion

Comparable reasoning patterns exist for inverse proportion. Consider a container of colored liquid inside a right triangle where the triangle can be tilted and the water levels on the left and right side can be measured on a built-in scale. We call this a “water triangle.”

Problem Statement and Typical Solutions

Rotate your water triangle until you get a measurement of 4 units on the left side and 6 units on the right side.

Suppose the triangle is tilted even more until the water level on the right side is at 8 units. Predict what the water level in units will be on the left side.

Someone with knowledge about the area of triangles might reason: “Initially the area of the water forming the triangle is 12 since ½ * 4 * 6 = 12. The amount of water doesn’t change so the area won’t change. So the answer is 3 because ½ * 3 * 8 = 12.”

A correct multiplicative answer is relatively rare. By far the most common answer is something like: “2 units because the water level on the right side increased by two units so the water level on the left side must decrease by two units and 4 – 2 = 2.” Less frequently the reason for two units is: “Before there is a total of 10 units because 4 + 6 = 10. The total number of units must stay the same so the answer is 2 because 2 + 8 = 10.”

So again we see individuals who are not at the formal operational level apply an additive strategy to solve an inverse proportion rather than a multiplicative strategy. And, like the direct proportion, this incorrect strategy appears to be logical to the individual and appears to give a reasonable answer. Students are very surprised when they actually carry out the experiment and tilt the triangle to find the answer is 3 and not 2 as they so confidently predicted.

Viewing These Strategies as Functional Relations

Let T be the height of Mr. Tall and S be the height of Mr. Short, then the correct multiplicative strategy can be expressed as T/S = 3/2; this is a constant ratio relation. The incorrect additive strategy can be expressed as T – S = 2; this is a constant difference relation. Here is the graph for these two equations. For the numeric values involved in the problem statement, these graphs are “similar” and it is easy to see why individuals consider their incorrect answers perfectly reasonable.

Now consider our inverse proportion using the “water triangle.” Let L be the height of the water on the left side and R be the height of the water on the right side, then the correct multiplicative strategy can be expressed as L * R = 24; this is a constant product relation. The incorrect additive strategy can be expressed as L + R = 10; this is a constant sum relation. Here is the graph for these two equations. For the numeric values involved in the problem statement, these graphs are “similar” and it is easy to see why individuals consider their incorrect answers perfectly reasonable.