Proportional Reasoning and Multiplicative Thinking
Proportional reasoning requires the capacity to see situations of comparison in a multiplicative rather than additive sense. Consider the numbers 2 and 10. When asked to describe what has happened to the number 2 to change it to the number 10, an additive thinker will state that it has increased by 8. A multiplicative thinker will also see the change in this way, but will also see the change as the number 2 being multiplied by 5.

Being able to make comparisons in additive and multiplicative terms is also referred to as absolute and relative thinking respectively. Susan Lamon (2006) provided a detailed example of how situations of comparison can be seen in absolute and relative terms. Her example is presented here, however the objects (animals) of comparison have been changed to crocodiles. Images have also been included to highlight how situations can be considered in additive and multiplicative terms, and the knowledge of numbers that is required for multiplicative thinking.
At the zoo, there are two longterm resident crocodiles that have affectionately been named Prickles and Tiny. When they arrived at the zoo, Prickles was 4 metres long and Tiny was 5 metres long. Five years later, both crocodiles are now fully grown. Prickles is 7 metres long and Tiny is 8 metres long
At the zoo, there are two longterm resident crocodiles that have affectionately been named Prickles and Tiny. When they arrived at the zoo, Prickles was 4 metres long and Tiny was 5 metres long. Five years later, both crocodiles are now fully grown. Prickles is 7 metres long and Tiny is 8 metres long
Which crocodile has grown the most? From an additive (absolute) perspective, both crocodiles have grown by the same amount of 3 metres. From a multiplicative (relative) perspective, Prickles has grown 3⁄4 of his beginning length and Tiny has grown 3/5 of his beginning length. Prickles has grown more than Tiny because 3/4 is greater than 3/5.
Another diagram can show the relative change more directly. Placing Prickles on a number line, we can see his growth from 4 metres to 7 metres:
Another diagram can show the relative change more directly. Placing Prickles on a number line, we can see his growth from 4 metres to 7 metres:
A similar diagram would show the 3/5 increase in Tiny’s growth, hence meaning that he is now 1 and threefifths his original size. If he had doubled, he would have been 10 metres. Tiny had a 60% increase, which means that he is now 160% of his original size. Being able to see these various equivalent relationships is a key to proportional reasoning.
Sometimes situations that appear to be proportional and therefore multiplicative, are actually additive. Cramer, Post and Currier (1992, p159) described an additive situation that was mistakenly identified as multiplicative, as follows:
Two cyclists, Harry and Freda, are cycling equally fast around the cycling track. Freda commenced cycling before Harry arrived at the track and had completed 9 laps when Harry had completed 3. When Freda has completed 15 laps, how many laps will Harry have completed?
Cramer, et al. reported that when presented with such situations, many students (and adults) regard them as proportion situations, and in the example above, would calculate the solution to be 45 (rather than 9). This is an indication of how often people become accustomed to ‘school problems’ and automatically apply learned procedures, sometimes inappropriately.
As exemplified in the above examples, proportional reasoning can be tricky. Here is another situation that requires some thinking to determine whether it is proportional or not. It is a question that Susan Lamon also included in her research:
References
Cramer, K., Post, T. and Currier, S. (1992). Learning and teaching ratio and proportion: Research implications. In D. T. Owens (ed.), Research Ideas for the Classroom: Middle Grade Mathematics. Ma cmillan, New York, pp. 159178.
Lamon, S. (2006). Teaching fractions and ratios for understanding. Mahwah: Erlbaum.
Sometimes situations that appear to be proportional and therefore multiplicative, are actually additive. Cramer, Post and Currier (1992, p159) described an additive situation that was mistakenly identified as multiplicative, as follows:
Two cyclists, Harry and Freda, are cycling equally fast around the cycling track. Freda commenced cycling before Harry arrived at the track and had completed 9 laps when Harry had completed 3. When Freda has completed 15 laps, how many laps will Harry have completed?
Cramer, et al. reported that when presented with such situations, many students (and adults) regard them as proportion situations, and in the example above, would calculate the solution to be 45 (rather than 9). This is an indication of how often people become accustomed to ‘school problems’ and automatically apply learned procedures, sometimes inappropriately.
As exemplified in the above examples, proportional reasoning can be tricky. Here is another situation that requires some thinking to determine whether it is proportional or not. It is a question that Susan Lamon also included in her research:
 Can you think of a time when you were half your mother’s age?
 Can you think of another time when you were half your mother’s age?
 Why not?
References
Cramer, K., Post, T. and Currier, S. (1992). Learning and teaching ratio and proportion: Research implications. In D. T. Owens (ed.), Research Ideas for the Classroom: Middle Grade Mathematics. Ma cmillan, New York, pp. 159178.
Lamon, S. (2006). Teaching fractions and ratios for understanding. Mahwah: Erlbaum.