Proportional Reasoning
Proportional reasoning involves thinking about relationships and making comparisons of quantities or values. In the words of John Van de Walle, “Proportional reasoning is difficult to define. It is not something that you either can or cannot do but is developed over time through reasoning ... It is the ability to think about and compare multiplicative relationships between quantities” (2006, p. 154).
Multiplicative Thinking
This concept involves reasoning about several ideas or quantities simultaneously. It requires thinking about situations in relative rather than absolute terms. Consider the following problem. If one dog grows from 5 kilograms to 8 kilograms and another dog grows from 3 kg to 6 kg, which dog grew more? When a student is thinking in absolute terms or additively, she/he might answer that both dogs grew by the same amount. When a student is thinking in relative terms, she/he might argue that the second dog grew more since he doubled his previous weight, unlike the first dog who would have needed to be 10 kg to grow by the same relative amount. While both answers are viable, it is the relative (multiplicative thinking) that is necessary for proportional reasoning.
Why is this important?
Helping students bridge from additive to multiplicative thinking is complex but starts early. It forms the backbone of the mathematics curriculum and includes important and interconnected ideas such as multiplication, division, fractions, decimals, ratios, percentages and linear functions. It requires time, a variety of situations and opportunities to construct their understanding in multiple ways.
PAYING ATTENTION TO K–12 PROPORTIONAL REASONING
Support Document for Paying Attention to Mathematical Education
Multiplicative Thinking
This concept involves reasoning about several ideas or quantities simultaneously. It requires thinking about situations in relative rather than absolute terms. Consider the following problem. If one dog grows from 5 kilograms to 8 kilograms and another dog grows from 3 kg to 6 kg, which dog grew more? When a student is thinking in absolute terms or additively, she/he might answer that both dogs grew by the same amount. When a student is thinking in relative terms, she/he might argue that the second dog grew more since he doubled his previous weight, unlike the first dog who would have needed to be 10 kg to grow by the same relative amount. While both answers are viable, it is the relative (multiplicative thinking) that is necessary for proportional reasoning.
Why is this important?
Helping students bridge from additive to multiplicative thinking is complex but starts early. It forms the backbone of the mathematics curriculum and includes important and interconnected ideas such as multiplication, division, fractions, decimals, ratios, percentages and linear functions. It requires time, a variety of situations and opportunities to construct their understanding in multiple ways.
PAYING ATTENTION TO K–12 PROPORTIONAL REASONING
Support Document for Paying Attention to Mathematical Education