## Trees, Graphs and the Leaving Certificate

I’m starting to get the hang of some of the differences between things here in Ireland and the United Kingdom, both domestically and in the world of work.

One of the most important points of variation that concerns academic life is the school system students go through before going to University. In the system operating in England and Wales the standard qualification for entry is the GCE A-level. Most students take A-levels in three subjects, which gives them a relatively narrow focus although the range of subjects to choose from is rather large. In Ireland the standard qualification is the Leaving Certificate, which comprises a minimum of six subjects, giving students a broader range of knowledge at the sacrifice (perhaps) of a certain amount of depth; it has been decreed for entry into this system that an Irish Leaving Certificate counts as about 2/3 of an A-level for admissions purposes, so Irish students do the equivalent of at least four A-levels, and many do more than this.

There’s a lot to be said for the increased breadth of subjects undertaken for the leaving certificate, but I have no direct experience of teaching first-year university students here yet so I can’t comment on their level of preparedness.

Coincidentally, though, one of the first emails I received this week referred to a consultation about proposed changes to the Leaving Certificate in Applied Mathematics. Not knowing much about the old syllabus, I didn’t feel there was much I could add but I had a look at the new one and was surprised to see a whole `Strand’, on Mathematical Modelling with netwworks and graphs.

In this strand students learn about networks or graphs as mathematical models which can be used to investigate a wide range of real-world problems. They learn about graphs and adjacency matrices and how useful these are in solving problems. They are given further opportunity to consolidate their understanding that mathematical ideas can be represented in multiple ways. They are introduced to dynamic programming as a quantitative analysis technique used to solve large, complex problems that involve the need to make a sequence of decisions. As they progress in their understanding they will explore and appreciate the use of algorithms in problem solving as well as considering some of the wider issues involved with the use of such techniques.

Among the specific topics listed you will find:

• Minimal Spanning trees applied to problems involving optimising networks and algorithms associated with finding these (Kruskal, Prim);
• Bellman’s Optimality Principal to find the shortest paths in a weighted directed network, and to be able to formulate the process algebraically;
•  Dijkstra’s algorithm to find shortest paths in a weighted directed network; etc.

For the record I should say that I’ve actually used Minimal Spanning Trees in a research context (see, e.g., this paper) and have read (and still have) a number of books on graph theory, which I find a truly fascinating subject. It seems to me that the topics all listed above  are all interesting and they’re all useful in a range of contexts, but they do seem rather advanced topics to me for a pre-university student and will be unfamiliar to a great many potential teachers of Applied Mathematics too. It may turn out, therefore, that the students will end up getting a very superficial knowledge of this very trendy subject, when they would actually be better off getting a more solid basis in more traditional mathematical methods  so I wonder what the reaction will be to this proposal!

### 18 Responses to “Trees, Graphs and the Leaving Certificate”

1. […] “I’m starting to get the hang of some of the differences between things here in Ireland and the United Kingdom, both domestically and in the world of work. One of the most important points of variation that concerns academic life is the school system students go through before going to University …” (more) […]

2. There is of course no such thing as the ‘UK system’ when it comes to education. Scotland, in particular, has a very different system and does not use A-levels. Sorry for being pedantic, but….

• telescoper Says:

Quite right. I meant England and Wales. I have amended accordingly.

• telescoper Says:

I think a united Ireland is a possibility now (which it has never been until now). I wouldn’t put the odds better than 50-50 right now, but there are many things that could happen that would change those odds considerably on the 10-year timescale you mention.

I think independence for Scotland is less likely, but you never know.

3. p.s. welcome to Ireland. It’s a great wee place.

4. John Peacock Says:

Well, this brings back memories. When I was at school, it was the early days of the SMP (Schools Mathematics Project, alias “modern maths”). This was a radical rewriting of the maths syllabus to help prepare kids for how the subject was taught in universities (interesting that this was allowed in days when only <10% of kids would go to university). So pretty much the first thing I did at secondary school, aged 11, was matrices and linear programming. I imagine the maths teachers must have blown a few fuses when this was introduced. But the staff at my school were canny: once O-levels were over, they basically said "OK, you've been messing around with fancy concepts for 5 years, now you're going to spend the first 3 months of A-level doing nothing but practice differentiation and integration". So I ended up with a great mixture of technical drilling, but on a foundation where big and general ideas were put in at an age where you just soaked them up. But it seems this revolution couldn't be sustained, and things are more traditional in the various parts of the UK now.

Pushing through revolutions is hard. During my PhD I witnessed carnage in Cambridge undergraduate 1st-year maths for physics. A new lecturer took the course over, decided all this div grad and curl stuff was hopelessly old-fashioned, and completely rewrote it in terms of differential geometry. Trouble was, all the tutors who had to support it had received the traditional education themselves, and mostly had no idea what a one-form was. So the brave innovator was sacked and things returned to normal. But he had a point, and the maths we deliver to our physics students remains as outdated as ever.

5. Isn’t education devolved in N Ireland and Wales as well as Scotland, so what A-levels mean in NI and Wales may also diverge from England?

Phillip Helbig’s comment about NI joining Ireland on a shortish timescale brings back Churchill’s line about the issues around “the dreary spires of Fermanagh and Tyrone” still being there even after the end of WWIiI. Unionism in NI (and Scotland) is still a powerful force – look at the DUP!

• I don’t know about Northern Ireland, but A-levels are treated the same in Wales as in England.

• Thought Welsh students already had the option of eg Welsh Bacc at A-level?

• It’s an option, but doesn’t alter how A-levels are interpreted. Many students in Welsh universities are from England.

6. In fact do the NI “troubles” warn against the dangers of unlimited immigration, in this case mainly of Scots?

7. When I did my A-levels in 1998 one of the module choices for the Further Maths course (at A or AS-level) was called Decision Maths. From what I recall it was very similar in content to the proposed Mathematical Modelling with networks and graphs. I’m not sure if it’s still available as an option, but it was interesting and refreshingly different from the other maths modules. That said, it led to my worst exam result, but mainly because the nature of it meant there wasn’t enough time to get through all the questions. (In reality I should have taken a statistics module, which would actually have been far more useful!)

8. I think the Irish system is quite good, and similar to most continental countries. With a good grounding in physics or maths, there’s no reason you can’t catch up in 1st year uni.
In our many sojourns abroad with Dad, our family often noticed that English scientists rarely seemed to have a good working knowledge of French or German. I presume this has something to do with concentrating on three subjects for A-levels? More recently, I notice that many scholars at Cambridge interested in Einstein’s original works don’t speak German. It gives our group an edge- I’m no expert, but I can read well enough to recognise something that’s worth taking a closer look at!

9. Greg Metcalfe Says:

It have been influenced by the boom in graph usage in social networking in the Internet giants, such as Facebook. More generally, it’s also showing up more in high performance technical computing, where drives toward exascale computing (in US national labs and elsewhere) have recently moved quite a bit in the direction of “Big Data” (I detest that term) and graphs than previously. The national labs exist in part to support industry, but also to support the US defense industry. One intersection of these two purposes is, unfortunately, surveillance at scale.