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.…

Chris Rasmussen, chris.rasmussen@sdsu.edu, maa.org/cspcc

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

There are 4 regular  international conferences on undergraduate mathematics:

RUME (USA), DELTA (Southern hemisphere), one in the UK, one in Europe

How do you make sense of Ax = b where x and b are vectors? (Larson & Zandieh, 2013). There are multiple representations:

• Linear combination, vector addition
• System of equations, intersection of 3D lines
• Transformation, one vector mapped to another

Chalk Talk practices include

• Verbalising everything written on the board
• Board choreography
• Using pointing gestures to highlight key issues, and connections
• Using rhetorical questions to signal transitions, reflection or to check for understanding

Students in lecture-oriented classes has 1.5 times the chance of failing compared to inquiry-oriented classes. Inquiry-based learning is especially good for women. The continued use of traditional lecturing without some form of active learning can be seen as a way to limit the success of many students. …

The exploitation of videos in teacher training

Marcia Jussara Hepp Rehfaldt, University of Vale do Taquari, Brazil

5^th year of a research project on primary mathematics education. Using videos to teach mathematical modelling to teachers in training.

Steps for creating videos

1. Mathematical modelling practices were developed by researchers and recorded on video, e.g. how to choose a team.
2. Editing the image and created slides on an online platform
3. The narrative and final editing were carried out.

The narrative was tabulated. Each of 14 screens was described by what was said and displayed.

What was done with the video?

• Training courses with undergraduate students in teaching
• Teachers in 6 schools
• Online training courses – no professors wanted to go to the northeast state of Brazil after kidnappings

After the video, sent questionnaire to participants and analysed using Tagcrowd software to find most mentioned words (word cloud).

• Common words: mathematica, tendancias, contribuiu, video, clara, ensino, aprendizagem, alunos.

A novel approach to mathematics examination design and marking

David Easdown (University of Sydney, Australia), Ruth Corran (American University of Paris, France), Brad Roberts (University of Sydney)

Presentation at the 11th Southern Hemisphere Conference on the Teaching and Learning of Undergraduate Mathematics and Statistics in Gramado, Brazil, November 2017, “Brazil Delta 2017’

The new approach to marking evolved over 3-4 years, prompted by policy changes regarding assessment.

Old method of assigning grades: norm-referenced: use professional judgement to decide what a passing grade was, map to the students in the class. Policy said a maximum of 5% can achieve the top bracket grade (over 85%).

Intermediate method: Instead of grades, assessment classifies students’ test performance as ‘masterful’, ‘excellent’, ‘credible’ or ‘routine.’ Less greyness around borderlines compared to previous norm-referenced scale.

Policy change meant the raw mark achieved on an assessment would be their score. Tests need to be developed so that a student who deserved to pass would score 50% of the marks on the test, but only those who deserved a distinction would achieve the marks.…