Understanding of limits and differentiation as threshold concepts in a first year mathematics course

Greg Oates (Robyn Reaburn, Michael Brideson, Kumudini Dharmasada)

Presentation at the 11th Southern Hemisphere Conference on the Teaching and Learning of Undergraduate Mathematics and Statistics (‘Brazil Delta 2017 for short)

Misconception from school: a graph can never cross an asymptote (Greg had never seen such a graph before university).

Beware of demonstrations that cement misconceptions like walking halfway to the wall repeatedly, the tortoise and the hare.

Threshold concepts – once you understand it, you can’t go back. Meyer & Land, 2003.

Bachelard 1938  -topics that are epistemological obstacles, hard to get.

Are limits threshold concepts?

Students see trickery when limits are taught.

Study

• 1^st year calculus
• Pre-survey
• Interviews
• Repeat survey in 2^nd semester
• In survey, responses were true, false, never seen before, forgot
• Only 14 responses from 270 students
• 4 said you can’t cross a vertical or horizontal asymptote
• 7 said the limit of sin x / x as x tends to 0 could probably be rewritten so the denominator is not l

In interviews, the way students talk about limits, asymptotes, infinity showed that it is a threshold concept.