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?**

The concept of algebra, that is, the concept that you can use imaginary variables (which we represent by letters or words) to hold some possible values for a thing, or all possible values for a thing.

It is quite hard to teach this concept to someone who does not intuitively ‘get it’ right away.

I would extend one of your concepts to also ‘knowing what a proof is and what it means’. I’ve encountered this threshold tutoring secondary school students.

I remember in secondary school, everyone else in my (advanced maths) class got matrix multiplication within a few sessions, whereas it took me over a week as the teacher didn’t give us a rigorous formal definition for it.

I highly doubt this is representative, but for me personally, understanding how to solve quadratic equations by completing the square. It took me quite a long time to understand it (back then I was a kid, self-teaching with my big brother’s old math book). Once I did, there was a kind of “click”, and from there I was able to understand most of elementary calculus with no further trouble 😛

I am not sure what the definition of “threshold concept” is, I rather doubt there is one. And even if there is then “knowing when a proof is convincing enough” will not be one of them. I still regularly come up with “proofs” that turn out not to be convincing.

Of course one problem here is that for a mathematician a definition is actually a test which allows

us to identify which snarks are boojums. The best definitions are those that are easiest to use.

This means that mathematicians are not using the term in the way it is usually used. And to complicate the issue, questions of the form “What is a vector?” rarely have helpful answers. The right question is “What can I do to a vector?”. (One way of expressing this is that all mathematicians think categorically, whatever their private opinion of category theory might be.) Of course in teaching it might still be wise to address the “What is?” question first.

Note the problems I am pointing out here arise even in discussions with physicists, and physics might be thought of as a closely related subject.

As a minor point, the problem with the directed line segment approach to vectors is that you

are actually working in the tangent bundle to the real plane. You can hide this by declaring

a vector to be an equivalence class (orbit under translations), but this simply means that the

problems change their hiding place.

(I look forward to arguing about these things in Gent.)

The general idea is that a threshold concept is something that proves difficult initially, yet is clear in hindsight and for which mastery gives the student a qualitatively improved understanding of the subject.

In first year linear algebra (which is the current cross I have to bear), the concept of “linear independence” seems to meet these requirements. Many students struggle for weeks with the definition, but then suddenly (for the lucky ones) “the penny drops” and it all seems totally obvious. Once they have mastered linear independence, suddenly finding bases and calculating dimensions becomes easy and natural.

On the other hand, calculating the determinant of a matrix is also something that students struggle with, but it is not a threshold concept. It’s just memorizing the rules and the signs that some students find difficult. Mastering it doesn’t really lead them anywhere except for being able to calculate the determinant of a matrix.

The trouble with our existing teaching materials (for my first year unit) is that the time and number of pages spent on defining linear independence and calculating determinants is almost exactly the same. Each of them are simply defined and a couple of examples given. No student would have a clue (unless they were told) that one is a peripheral computational technique and the other is a critical conceptual issue.

The aim is to spend more time in the future on the bottlenecks and problem areas and less on the routine computations…

Of course whether it works or not is a story for next year’s blog posts…

It seems obvious that we should focus on the stuff that causes difficulties and will be needed later, is a “threshold concept” more than this?

To take your example of the determinant. Suppose that in the rest of the course you were going to make substantial use of the determinant, say to work with Pfaffians and derive results about perfect matchings. Then perhaps in this case their difficulties would not be peripheral, and computing determinants would be a threshold concept?

I vote for the equivalence relation theorem.

This is the difference between convincing students that something is true (using in part your authority as teacher) and having them understand WHY it is true, at least in some vague sense.

Many students can plow through inductive proof after inductive proof using the recipe they’re taught. Far fewer have an intuitive notion of what induction is, and why the proofs are structured the way they are.

Similarly in Linear Algebra there are many more students who can properly perform row reductions and find the nullspace of a matrix without any notion of what they’re doing.

As far as examples of actual concepts, off the top of my head (some mentioned already):

– “derivative is the slope of the curve”

– “what is a function”

– “matrices linear transformations”

– “linear independence”

Even further back, the idea of ‘doing the same thing to both sides of the equation’ idea. Some kids get this at an early age and can start playing with equations freely. Others are apprehensive and then they start to fear mistakes, start to fear manipulating expressions, and ultimately start to fear maths.

The problem here is that breeds of mathematicians have different perspectives on what something is, so sometimes we ought to ask what the “concept” at the heart of the matter is. Take for example the humble idea of a vector (as above). I agree and disagree with Chris, and probably at cross-purposes anyway. The “definition” of a vector is one thing, and knowing what it is useful for is another. I would think that a pure mathematician and applied mathematician might have trouble comparing what they believe is difficult about vectors, but they would make progress if they unpacked the difficulty and dispensed with the higher-level notion of a vector altogether. The pure mathematician thinks of a “coordinate vector”, a position in a multidimensional space. The idea of “direction” may not really be at the fore. For a physicist, the vector is a geometric entity encapsulating magnitude and direction, and knowing how to take scalar and cross products is an important part of “using” vectors. But these may not be the threshold concepts.

For the pure mathematician the threshold concept might be the idea that a single mathematical object can have components, it can represent more than one thing. For the physicist, perhaps there is a geometric threshold concept underlying vector arithmetic which is more important.

Peter came up with a cracker: the equivalence relation theorem. OMG this is tough to teach, but why? Linear independence, as Gordon pointed out, is a good example of a TC as it often has that “penny dropped” phenomenon. And I agree totally with Brian; induction can easily be swept under the carpet without ever scratching the surface of its true meaning.

3 apples + 2 apples “is” an addition. 3 apples + 2 bananas is a (linear) combination, AKA a vector.

Then, 5 cents/apple is a co-vector and (3 apples, 2 bananas) @ [5 cents/apple, 4 cents/banana] = 23 cents (in the underlying scalar field).

Sometimes called “fruit salad arithmetic”, sometimes called Linear Algebra.

Regards

–schremmer