There are methods by which teachers can guide students in the correct application of proportional reasoning.
Robert Karplus, a science educator in the 1960s and 1970s, investigated all these forms of reasoning in adolescents & adults.
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.
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.
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.
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.
Also the additive strategies noted above cannot simply be labeled as "incorrect" since they do correctly match other real world situations.
It is critically important that students on their own recognize that their current mode of reasoning, say that it is additive, is inappropriate for a multiplicative problem they are trying to solve.
Tall in paper clips, the measuring tools can be introduced and the students can test their strategies.
Tall is actually nine paper clips high and this will set up some cognitive dissonance.
This is a prime time for the teacher to move the lesson into the second stage of the learning cycle.
In all the experiments noted above there are two variables whose values change based on a fixed relation.
This can also be achieved by exploring the other extreme where the original picture is blown up to poster size and the daughter is 100 cm high.
Once cognitive dissonance is present, the teacher can introduce the correct relation, constant ratio.
Most students, including those still at the concrete operational stage, will quickly answer that the father's height must also double.
For inverse relations, such as the "water triangle", limiting cases can also introduce cognitive dissonance.
Students will abandon the additive strategy at this point realizing that 0 cannot be the correct answer.
Again, consider Newton's equation for the force of gravity: If a student understands the functional relation between the variables, then he/she should be able to answer the following thought experiments.
What would happen to the force of gravitational attraction if: Generally, thought experiments must be confirmed by experimental results.