In recent years, science education has been taking note of the idea of ‘threshold concepts’. The idea came out of studies in how students learn economics by Erik Meyer and Ray Land, but has much wider application.

We’ve done a bit of study in this at Waikato, particularly for electronics – see for example Jonathan Scott and Ann Harlow (N.B. Full references are listed at the end.)

The basic idea is that some concepts in a subject are by their nature exceedingly difficult to learn. That’s because they typically require a change in the way a student thinks about something. They’re not just another fact to remember, or another technique for solving a problem, but transform the whole approach a student will take to something. With a threshold concept, there’s no ‘going-back’; once you have grasped it, the way you do things will change. Since students typically build knowledge on the foundation of what they already understand (or believe they understand) fitting in something that involves a shift in their fundamental thinking isn’t easy. A threshold concept may need to be presented to students many times before it gets grasped. The first time students are presented with it, it is likely to result in a sea of perplexed faces.

It’s tricky to give an example of a threshold concept for a general audience, because many are very specific to a discipline. Thinking back to childhood, one might recognize an example in understanding where money comes from. A very young child doesn’t get it – money is simply something that comes from Mum’s purse that can be exchanged for useful stuff. When Mum says "No, we don’t can’t afford it", the child is perplexed. "But there’s money in your purse", he thinks. "You can always go to the bank and get some more." At some point he learns that there isn’t an endless supply of it – it gets earned – even the stuff that comes from the ATM – and at that point his thinking about it changes.

I ran into a threshold concept in a lecture on Friday. This one pertains to units in physics. In particle physics, one often talks about the mass of elementary particles as being in the unit MeV/c2. (N.B. It’s tricky to write equations in MoveableType – the ‘2’ here means ‘squared’.) For example, the mass of an electron is 0.511 MeV/c2.

What does this mean? Well, MeV (mega electron volt, or a million electron volts) is a measure of energy. One electron volt is the energy that an electron acquires when it is accelerated through a potential difference of 1 volt. It’s equal to 1.6 times 10 to the power of -19 joules. Students don’t typically have a problem with this – it’s just another unit – in the same way one can measure speed in metres per second or kilometres an hour or miles per hour, one can measure energy in joules (the S.I. unit of energy) or electron volts, (or kcal for that matter).

Now, recall Einstein’s famous E=mc2 equation, relating the rest-mass energy E of a particle to its mass m. Divide both sides by c2, and we get E/c2 = m. That is, an energy divided by a velocity squared is a mass. Students are with me so far.

But now the threshold. I say we can write mass as so-many MeV/c2 (a MeV divided by the speed of light squared.) This they don’t like. Why? Because it involves the speed-of-light as part of the unit. The trouble is grasping what MeV/c2 represents. It’s sort of an equation but not quite. I think the problem here lies with the fact that my students don’t fully understand units in general – they have the idea they are a tag-on at the end of a number, rather than something that is absolutely integral to the quantity itself. What’s curious was that if I wrote the mass of an electron as (0.511 MeV)/c2, they were happy. It’s an algebraic equation – take the 0.511 MeV, and divide it by the value for the speed of light squared. But shift the brackets to 0.511 (MeV/c2) and suddenly it’s a conceptual jump too far.

It certainly appears to be a threshold for them, and, according to the literature, I can’t expect them to grasp it quickly. Thinking back to my student days, I didn’t grasp it quickly, either.

References:

Land R, Cousin G, Meyer JHF and Davies P (2005). Threshold concepts and troublesome knowledge (3): Implications for course design and evaluation. In Rust C (ed.) Improving Student Learning – equality and diversity. Oxford: OCSLD.

Meyer JHF and Land R (2003). Threshold concepts and troublesome knowledge (I): linkages to ways of thinking and practising. In Rust C (ed.), Improving Student Learning – ten years on. Oxford:OCSLD.

Meyer JHF, Land R and Davies P (2006). Implications of threshold concepts for course desing and evalutation. In Meyer JHF and Land R (eds.) Overcoming Barriers to Student Understanding: threshold comcepts and troublesome knowledge. London and New York:Routledge.

Scott J, Harlow A, Peter M and Cowie B. (2010). Threshold Comcepts and Introductory Electronics. Proceedings of Australasian Association for Engineering Education, Sydney.