Fractions can quite literally serve as a puzzling component of mathematics.

One of the reasons fractions are perceived to be difficult is what members of the math community call gap reasoning.

Gap reasoning can limit students’ understanding of the problem. The method leads students to assess distances as equal by appearance over measurements.

In their paper “Escaping the gap,” Dr. Patrick Sullivan and Professor Joann Barnett use an original classroom activity to establish gap reasoning as an ineffective comparison strategy.

About their research

Sullivan and Barnett investigated elementary students’ use of gap reasoning.

From early education, students often struggle to make fraction comparisons using valid reasoning. Students may reach correct answers using gap reasoning, but for the wrong reasons.

For example, when comparing 1/3 and 1/4, a student who uses gap reasoning would claim that 1/3 is greater than 1/4 because 1/3 is two away from the whole while 1/4 is three away from the whole.

While students would reach the correct answer following that logic, the same result would not apply to a comparison of 5/6 and 7/8. Students using gap reasoning would incorrectly identify the fractions as equal.

About their focus

Addressing errors in reasoning at the early stages of education can prevent problems with fractions from continuing into adulthood. Failure to understand fractions can often bring frustration to adults.

“People see it as the F word of elementary mathematics,” said Sullivan.

Sullivan and Barnett avoided focusing on students’ ability to reach right answers. Instead, they directed their attention to understanding students’ reasoning for each answer.

The results of their study allowed Sullivan and Barnett to bring awareness to the importance of asking students to share their reasoning from the early stages of the education process.

“It is critical that we, as educators and parents, ask our students the why to explore how students’ understanding of fraction concepts is developing,” said Sullivan.

Barnett’s work related to her master’s degree thesis, which served as the genesis of their research.