Today I attended a fabulous workshop on “Threshold Concepts in Engineering”, organised by Caroline Baillie and Sally Male, and it got me eager to jot down some ideas for pure maths. The idea of a threshold concept needs some pinning down, but it is vaguely something that our students need to master in order to progress to other ideas; a competent understanding of a threshold concept opens the door to many other concepts. We also think of it as something that is typically difficult for the student and is transformative. That is, once the student “gets it”, it can change their way of viewing previous notions, it could change the way they approach and do things, and it could change the way they see themselves as students of a particular discipline (i.e., a student of mathematics then regards themselves as a mathematician). See our faculty’s threshold concepts webpage for more.
But what about some examples in early tertiary mathematics education?
(By the way, the maths taught in secondary schools in Australia does not really get as far as what I would view as “real” mathematics. So before you tell me that your students learn measure theory in middle school, I know that some of the topics that I describe below may not convert to tertiary maths at your institution!)
Representations of objects which denote things other than numbers
One good example of this is the idea of a vector. It is often taught as “arrows” and we perform arithmetic on them by joining their ends or making parallelograms with them. I remember not really getting this concept straight away and wasn’t comfortable until the teacher told me it was an -tuple. (Pure mathematicians always want to know a clear definition of something!) Other examples include matrices, and denoting arbitrary matrices with a symbol like . We still have issues in first year where our students are fine with denoting a real number, but struggle with it denoting a matrix or a statement in logic.
The definition of something
I can’t think of a situation in secondary school where a teacher actually defines something. Can you? They seldom define trigonometric functions, but instead they “define them by association”. That is, the student learns a set of properties that the mathematical thing has, which turn out to define the object. I recently saw this concept come up in a tutorial. Almost every student last week could not solve the following problem:
Q: Show that if a square matrix satisfies , then is invertible and .
Most students started their reasoning with . The experienced reader will know that to show that a matrix is the inverse of a matrix , we need to apply the definition of “inverse”: . (So there is a simple one line answer to the above problem: ).
There are also some other concepts which are definitely threshold concepts such as continuity, linear independence and knowing when a proof is convincing enough.
Can you think of some others?