Blogging from The Tenth Southern Hemisphere Conference on the Teaching and Learning of Undergraduate Mathematics and Statistics

Dr Tracy Craig – UCT (photo taken from http://uct.academia.edu/TracyCraig)

Live blogging: Note that these are notes I’ve taken live, but will edit this today into a more readable format. I want to put this up straight away though to see if I have any obvious misunderstanding. Equations will also be put into more readable format ASAP.

Research with Trevor Cloete  – UCT Centre for research in engineering and science education.

Dynamics Education research group: With mechanical engineers and the Academic Development Programme.

Current study: Vector proficiency

Originally interested in the difficulties students find with (originally dynamics) and now vectors. In particular differences in notation and terminology between mathematics and dynamics (more purely engineering).

Vectors are used throughout dynamics.

Vectors are a very divided subject:

Some students ‘get’ vectors. Other students really struggle and have to memorise the process.

Can we find a threshold concept?

  • Run an assessment on vectors – lots of vectors, find the areas which differentiate those students that do and don’t get it.
  • Ran this on the first day of term in the dynamics class.
  • Will do it every semester.
  • Vector proficiency test: N=170
  • MCQ test, 29-31 items
  • Analysed using the Rasch measurement method
  • Problems ranked on axis of difficulty, matched against axis of student proficiency.

No clear clustering. However, some interesting things:

  • Items which were very easy: Find the dot product
  • Items which required the cross product: one was very difficult, other was less difficult.
  • Use of cross product in a context was moderately difficult.
  • Use of the dot product in a context was the most difficult. ie. finding the angle between two vectors.

There is currently no literature on the scalar product! This was a big surprise in the literature. There is some work on the cross product.

Focus on two questions:

Resolve two vectors into components parallel and orthogonal on two vectors (using bracket notation). Find the parallel component.

The other question used unit vector notation rather than bracket notation.

The questions were often paired like this. In general the unit vector notation was harder than the bracket notation.

Plot quartiles on a proficiency curve. Second and third quartiles doing better and worse respectively than expected, therefore students were guessing.

Only the best students generally chose the correct answer in the MCQ for one of the questions. The answer wasn’t the most popular answer for even the best students.

Types of error:

  • Carrying out vector product rather than a dot product
  • Non-useful diagrams (drawing 3d diagrams – unhelpful) – is it strange that we don’t want them to draw diagrams?
  • Using scalar product results incorrectly:
  1. Finding an angle or
  2. Turning result into a vector
  3. Or using moduli incorrectly
  • Weak basic arithmetic

Conclusions:

  • We argue that the geometric role of the scalar product is undestood weakly, it at all, by the majority of students.
  • The computational challenge of the vector product is greater than that of the scalar product, while using the vector product in a context is less challenging than using the scalar product in context.
  • No gut feeling about what the scalar product can give them.

Interesting that the vector product is computationally more difficult but there is no simple geometric interpretation.

How clear is this post?