Imagine that you are a raw potato…

Tuber or not tuber.

The words in the title of this post, surprisingly, are the first words in the Editors’ Preface to Land, Meyer and Smiths 2008 edited book “Threshold Concepts within the Disciplines”. Our group has been looking at the penetration of certain ideas through the discipline, examining how much the theory social constructivism accompanies the practice of group work for example, or, as in this case, seeing how many people identify threshold concepts in what they are trying to teach. Everyone who teaches first year Computer Science knows that some ideas seem to be sticking points and Meyer and Land’s two papers on “Threshold Concepts and Troublesome Knowledge” (2003 and 2005) provide a way of describing these sticking points by characterising why these particular aspects are hard – but also by identifying the benefits when someone actually gets it.

Threshold concept theory, in the words of Cousin, identifies the “the kind of complicated learner transitions learners undergo” and identifies portals that change the way that you think about a given discipline. This is deeply related to our goal of “Thinking as a discipline practitioner” because we must assume that a sound practitioner has passed through these portals and has transformed the way that they think in order to be able to practice correctly. Put simply, being a mathematician is more than plugging numbers into formulae.

As you can read, and I’ve mentioned in a previous post, threshold concepts are transformative, integrative, irreversible and (unfortunately) troublesome. Once you have passed through the hurdle then a new vista opens up before you but, my goodness, sometimes that’s a steep hurdle and, unsurprisingly, this is where many students fall.

The potato example in the preface describes the irreversible chemical process of cooking and how the way that we can use the potato changes at each stage. Potatoes, thankfully unaware, have no idea of what is going on nor can they oscillate on their pathway to transformation. Students, especially in the presence of the challenging, can and do oscillate on their transformational road. Anyone who teaches has seen this where we make great strides on one day and, the next, some of the progress ebbs away because a student has fallen back to a previous way of thinking. However, once we have really got the new concept to stick, then we can move forward on the basis of the new knowledge.

Threshold concepts can also be thought of as marking the boundary of areas within a discipline and, in this regard, have special interest to teachers and learners alike. Being able to subdivide knowledge into smaller sections to develop mastery that then allows further development makes the learning process easier to stage and scaffold. However, the looming and alien nature of the portal between sections introduces a range of problems that will apply to many of our students, so we have to ready to assist at these key points.

The book then provides a collection of chapters that discuss how these threshold concepts manifest inside different disciplines and in what forms the alien and troublesome nature can appear. It’s unsurprising again, for anyone teaching Computer Science or programming, that there are a large number of fundamental concepts in programming that are considered threshold concepts. These include the notion of program state, the collection of data that describes the information within a program. While state is an everyday concept (the light is on, the lift is on level 4), the concentration on state, the limitations and implications of manipulation and the new context raise this banal and everyday concepts into the threshold area. A large number of students can happily tell you which floor the lift is on, but cannot associate this physical state with the corresponding programmatic state in their own code.

Until students master some of these concepts, their questions will always appear facile, potentially ill-formed and (regrettably) may be interpreted as lazy. Flanagan and Smith raise an interesting point in that programming languages, which are written in pseudo-English with a precise but alien grammar, may be leading a linguistic problem, where the translation to a comprehensible form is one of the first threshold concepts that a student faces. As an example, consider this simple English set of instructions:

There are 10 apples in the basket.
Take each apple out of the basket, polish it, and place it in the sink.

Now let’s look at what the ‘take each apple’ instruction looks like in the C programming language.

for (int i  = 0; i < numberOfApples; i++) {
  // commands here

This is second nature to me to read but a number of you have just looked at that and gone ‘huh’? If you don’t learn what each piece does, understand its importance and can then actually produce it when asked then the risk is that you will just reproduce this template whenever I ask you to count apples. However, there are two situations that humans understand readily: “do something so many times” and “do something UNTIL something happens”. In programs we write these two cases differently – but it’s a linguistic distinction that, from Flanagan and Smith’s work “From Playing to Understanding”, correlates quite well with an ability to pick the more appropriate way of writing the program. If the language itself is the threshold, and for some students it certainly appears that it is, then we are not even able to assume that the students will reach the first stage of ‘local thresholds’ found within the subdomain itself, they are stuck on the outside reading a menu in a foreign language trying to work out if it says “this way to the toilet”.

Such linguistic thresholds will make students appear very, very slow and this is a problem. If you ask a student a question and the words make no sense in the way that you’re presenting them, then they will either not respond (if they have a choice) as they don’t know what you asked, they will answer a different question (by taking a stab at the meaning) or they will ask you what you mean. If someone asks you what you mean when, to you, the problem is very simple, we run the risk of throwing up a barrier between teacher and learner, the teacher assuming that the learner is stupid or lazy, the student assuming that the teacher either doesn’t know what they’re saying or doesn’t care about them.

I’ll write more on the implications of all of this tomorrow.

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