# Of Euler, webs and medical education

There is a book review of a biography of the Swiss mathematician Leonard Euler, ‘Leonard Euler: Mathematical Genius in the Enlightenment’ in the Economist. The article is titled ‘The Master of them all’, and quotes Pierre-Simon Laplace: ‘Read Euler, read Euler, he is the master of us all’. I only know a little of his achievements, but one bit to do with graph theory has always stuck in my mind when I think about medical education. Bear with me please, the metaphors are being stretched.

In basic statistics you quickly learn that numbers get very big when you think about permutations and combinations. All those factorial n! numbers. I liked the topic, because you can feel what is going on by simply going through the motions of counting — as long as your examples are not to big. After that, you need a formula.

Euler might be considered one of the earliest students of graph theory. From Wikipedia:

*The city of Königsberg, Prussia was set on the Pregel River, and included two large islands that were connected to each other and the mainland by seven bridges. The problem is to decide whether it is possible to follow a path that crosses each bridge exactly once and returns to the starting point. It is not possible: there is no Eulerian circuit. This solution is considered to be the first theorem of graph theory, specifically of planar graph theory*

Simple examples of what all this about are also in Wikipedia. If you have many nodes (vertices) and the lines between (edges), how many edges do you have? A simple example is shown below.

“10-simplex graph” by Koko90 – Own work. Licensed under CC BY-SA 3.0 via Commons.

There are lots of different sorts of structures, in terms of whether the edges are directional, whether circles are allowed etc, and also whether weightings are factored in. Most of this is beyond me. But for a subset of graphs, , you use formulas like n(n–1)/2 to describe the number of edges. The simple point is that as the number of nodes rises, the number of edges escalates faster than some of us might have imagined.

Now imagine, the nodes are people, like medical students, or staff who teach medical students. The edges represent communication between these individuals. Some communication is staff-staff, some student-student, and some staff-student. And the direction of flow can vary, some bilateral some more unilateral.

What has happened over time is that the number of nodes has increased. There are more students and more staff. The result is that when you double the nodes, the increased complexity is not merely double but rises in a polynomial way. This represents a major problem. For many medical schools even enumerating the number of staff who students are exposed to, is tricky. In my own unit, within a ten day attachment each one of 14 students might be exposed to a group of 15 teachers. If we keep the arithmetic simple, for each student, for 10 days, the number of combinations is 15!/10!5! The answer is over 3000 combinations. In reality there is structure to our timetabling, so the number is much smaller, but it remains large. And I have ignored the group effects of peer mixing or peer interaction.

Now think across the whole course, with ~ 300 students in each year, and a ‘teaching staff’, or staff to which students are exposed, of at least 300 [I have made this number up, but I suspect it is a conservative estimate].

Now some type of webs (because that it what they are) have a number of interesting properties. The web I am writing this on (as in the internet) was developed because information structures designed as webs have greater resilience than hierarchical command and control structures, that the military services traditionally liked (itself a type of ‘poor’ web). The internet grew out of work in the 1950s pointing out that if the Soviets attacked the US, a diffuse communication net, rather than one with giant dominant node or a few directional edges, was more robust.

I think you can also view much research in this way. Nodes are individual scientists, and we think that by making links across the work of may other scientists real advance is more likely (you can think of the nodes are persons, or microfields of enquiry). It is this freedom that makes many of us believe that much central control is not efficient for the generation of new ideas or any novel combination of ideas.

There are however downsides. In complex structures, communication across all the nodes comes with a cost. Just think of email. Most days my university seems to email me telling me that certain services I never use are not going to be available for a period of time. And then they email me to say then are up and running again. In a more hierarchical structure, some of this wasteful communication can be avoided (but with a cost).

In many situations people do not like the potential openness of webs, rather they want to control any possible flow of information, so they like knowledge to reside in silos, with few links between them, so that only they can control the information flow between these silos (and with links being directional). I think of these administrative measures as ‘pinching’ the network producing control points. If you have enough of them, you no longer have a useful web in the way the WWW is, but a standard hierarchical command and control web structure.

The resilience of some webs can be a downside under some conditions. The advantage of of simple hierarchical structures is that change can be controlled or even imposed (to some degree at least). If you have a teaching staff of >300 and 1500 students, changing direction in a complete web is almost impossible. Resilience to attack comes with the downside of resistance to change. This has always seemed to me to be one of the reasons that the pace of change in medical education is so slow. We have a large n, and the number of links large. Change is difficult.

There are seem approaches that historically are useful. First, we don’t attempt to link every road with every road in any country. We have trunk routes, and then smaller roads: we prune the branches. We also know that if you want to produce change, you are often better floating free of any large web — exocytosis if you like — to produce the classic skunkworks project. Companies are aware that this if often a sensible strategy but Deans or professional administrators hate the idea, because they think they know best. For any sense of creation, the mere burden of all these ‘routine’ information flows, stifles time for thinking; let alone thinking about something new. Remember Jim Watson’s comments about how his and Francis Crick’s pigeon holes were usually empty: they got no mail, when they were both in their most creative period.

Research — at least in good places — can almost institutionalise the skunkworks approach. This is simply because the goal of research, is research, although this statement appears to challenge many. There is no day job which has to be kept running. I once worked in a very large research institution in which multiple teams of researchers were trying to crack the very same problem, using different approaches.

For teaching, few similar opportunities exist. If you start a new course or even say a new medical school, you might be living the skunkworks for a short while, but eventually business as usual approaches will intervene. In medical education, more change — for good and bad it seems to me — has come from such new institutions. But eventually any successful skunkworks project will have to create its own trunk routes for information flow. And here is the fundamental problem: the structures we need to deliver teaching or research, are very different to the structures we need to generate novelty. For research we can sort of work within the system (or at least, some of us once could), because we once new how to finance great science. For teaching, in my experience things are much worse. Novelty is hard to finance, and the necessity for skunkworks less appreciated. My suspicion is therefore that much change will come from outwith the universities themselves.