Proportional reasoning is the ability to understand and reason about multiplicative relationships between quantities, such as ratios, rates, fractions, and scale factors. It involves recognizing how one quantity changes in relation to another and coordinating these changes as a consistent relationship.
In mathematics and physics In mathematics and in physics, proportionality is a mathematical relation between two quantities; it can be expressed as an equality of two ratios:
a b = c d {\displaystyle {\frac {a}{b}}={\frac {c}{d}}}
Functionally, proportionality can be a relationship between variables in a mathematical equation. For example, given the following equation for the force of gravity (according to Newton):
F = G m 1 m 2 r 2 {\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}}}
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.
Intellectual development In Piaget's theory of cognitive 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 Bärbel Inhelder formal operational reasoning takes many forms, including propositional reasoning, deductive logic, separation and control of variables, combinatorial reasoning, and proportional reasoning. Robert Karplus, a science educator in the 1960s and 1970s, investigated all these forms of reasoning in adolescents & adults. Mr. Tall-Mr.Short was one of his studies.
Examples
Inverse Proportion Comparable reasoning patterns exist for inverse proportion.
Water Triangle 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. This is called a "water triangle":
The water triangle is rotated until it shows 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.
Typical Solutions Someone with knowledge about the area of triangles might reason: "Initially the area of the water forming the triangle is 12 since 1/2 × 4 × 6 = 12. The amount of water doesn't change so the area won't change. So the answer is 3 because 1/2 × 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." Once again, some individuals who have not yet reached the formal operational stage tend to use an additive rather than a multiplicative strategy to solve problems involving inverse proportion. As with direct proportion, this incorrect strategy seems logical to the individual and appears to produce a reasonable answer. Students are often surprised when they perform the experiment and tilt the triangle, discovering that the correct answer is 3 rather than 2, as they had 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.
Teaching for Proportional Reasoning As any experienced teacher will attest, it is not sufficient to simply tell a student his/her answer is incorrect and then instruct the student to use the correct solution. The incorrect strategy has not been "unwired in the brain" and would re-emerge after the current lesson has been completed. Also the additive strategies noted above cannot simply be labeled as "incorrect" since they do correctly match other real world situations. For example, consider the following problem: On Independence Day this year Mr. Tall was 6 years old and Mr. Short was 4 years old. On a future Independence Day Mr. Short is 6 years old. How old will Mr. Tall be on that Independence Day? Similarly the constant sum relation can be correct for some situations. Consider the following problem. There are four beavers on the left side of a river and six beavers on the right side of the river. At a later time with the same group of beavers there are eight beavers on the right side of the river. How many beavers will there be on the left side? So there are situations where the additive relations (constant difference and constant sum) are correct and other situations where the multiplicative relations (constant ratio and constant product) are correct.
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