Reciprocal Thinking
 Courtney is in Year 8 and she is trying to interpret decimals in terms of fraction notation. She sees the decimal part as the denominator of a fraction, with larger denominators creating smaller fractions. This misconception is revealed when she chooses 0.3 as the larger of 0.3 and 0.4 (as 1/3 is larger than 1/4). Reciprocal thinking is one of the shorter-is-larger misconceptions.
Look at Courtney's answers to the Decimal Comparison Test. (You can double-click on the yellow notes for her reasons.)
Interviews with Courtney
 Talking About Place Value Video Images with text Making the Biggest and Smallest Numbers Video Images with text Number Between Video Images with text Hidden Numbers Video Images with text

See how Courtney would count from 1 to 2 with decimals and compare with other students.

Attempt a short decimal comparison test to check your understanding of Courtney's thinking. This will help you learn to diagnose this error.

Lesson ideas appropriate for students like Courtney.

Research on our Australian sample shows that 5% of students* use reciprocal thinking, but at one school, 27% of the Year 6 students were diagnosed this way.
* This percent includes both reciprocal and negative thinking, because the basic decimal comparison test does not separate these groups. Courtney's interview shows she is a reciprocal thinker. See full descriptions of reciprocal thinking and negative thinking.