Lessons taught; Lessons learnt

In the previous post (of what rapidly seems to be turning into a series) we explored the effect different scoring systems have on the placing of Canada, the USA, and Germany. The boundaries between different orderings looked like this:

These boundaries are a little complicated; it would be nice if they were something simpler, like straight lines.

The curved boundaries comes from the parameters used for our scoring system: is the value of silver relative to gold, and is the value of bronze relative to a silver. A country getting golds, silvers, and bronzes has total score . It's the in that score that caused the boundaries to be non-linear.

Let's change our parameters so that is the value of silver relative to gold (like before), and is now the value of bronze also relative to gold. This will make the score for each country .

What difference will this make to our graph? Before, the sensible parameter space was a square: could be anything between 0 and 1, as could . Now, can still vary between 0 and 1, but it doesn't make sense for to be greater than (we shouldn't value a bronze more than a silver). Our parameter space now is therefore going to be a triangle:

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

As before, move the blue point to alter the value of a silver and a bronze.

We can still see the four possible regions, corresponding to the four plausible orderings of the top three countries. The only two possible orderings which are not plausible are the two which put Germany first... but it wasn't that far off being a possibility. Indeed, it looks like all three of the lines cross just below our plausible region. Perhaps, if Germany had been able to convert some of its silvers into golds, or some of its bronzes into silvers, it actually could have claimed to be first under some weighting system.

This can be explored in the applet below, which lets you alter the number of golds, silvers and bronzes allocated to each country.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

[Source: 2010-top-three-places-linear.ggb.]

With this applet, we can experiment very quickly with lots of different scenarios (if you want to reset the tallies to the actual results, click the 'recycle' button in the top right). For example:

• With the same number of golds and silvers, Germany would have needed 12 bronzes (up from 7) for there to be a (quite small) region of silver/bronze weights which would put it first. In this case all six orderings would be possible, so we could legitimately argue that all three of the countries 'really' draw for first place.
• Changing one of Germany's bronzes into a silver also gives us a situation with a 'Germany wins' region. Interestingly, this is still true even if we reduce Germany's bronze tally all the way to 0.
• Similarly, changing one of Germany's silvers into a gold (giving a new gold tally of 11, with 12 silvers) gives us a 'Germany wins' region, even if we remove all Germany's bronzes. With 11 golds and 11 silvers, Germany would need 11 bronzes for a winning region, with 10 silvers 12 bronzes, with 9 silvers 13 bronzes,
  all the way down to needing 22 bronzes for a (very small!) winning region with 0 silvers.
• Moving over to Canada, it looks like we can increase the number of bronzes without limit, and still have a region where Canada come third. Giving Canada 9 silvers and 7 bronzes, we're in the interesting situation where Canada is always rated first or second, unless we count silvers as being worth the same as golds, where Germany and Canada come equal second/third, no matter how much we count a bronze as being worth.
• Giving Canada 10 silvers, and at least 13 bronzes, it will always be ranked first.

There are a lot of patterns and relationships in this that need explaining -- something, hopefully, we will do next time.

The 2010 Winter Olympics in Vancouver have been over for a few days now. 26 countries out of the 82 that competed won at least one of the 258 medals on offer. Toward the end of the event, quite a lot of interest was generated in which country was going to 'win', in the sense of coming top of the medals table. There's an interesting discrepancy in the medal table rankings around the world, though.

On the NBC Olympics site, and the official Vancouver 2010 site, the top of the medal table looks like this:

On the BBC and on Wikipedia, on the other hand, we see the top of the table looking like this:

The difference comes from a disagreement about the relative value of gold, silver and bronze medals. In the North American sites, all medals are valued equally, and the countries are sorted purely on the total number of medals won. The BBC and Wikipedia, on the other hand, use the standard IOC lexicographical ordering convention: sorting by golds, then by silvers, then by bronzes. This is the ranking system I've always been used to in Olympics, and indeed in any multi-event sporting occasion (such as the World Athletic Championships).

This difference in ranking systems first came to my attention during the 2008 Olympics, when the USA used their ranking system to claim victory over China (China won 51 golds to the USA's 36, but the USA won 110 medals overall, beating China's tally of 100).

One interesting mathematical question behind these tables is this: can we say anything about the rankings which is not sensitive to the particular relative values of gold, silver, and bronze? We might not be able to agree on whether the USA or Canada came first, for example, but perhaps it would always be the case that both of them always rank ahead of Norway. Also, is it possible to find a medal ranking system in which Germany comes first?

Simon Tatham has an interesting writeup of an investigation along these lines for the 2008 Olympic medal tables, which concludes with a wonderful diagram of the partial order of countries (using which we can see that, under any medal ranking system, Great Britain came fourth...). We could generate a similar partial order for the 2010 Winter Olympics.

Rather than doing this, I have created the following applet which will let you explore different medal ranking systems. Simply choose what percentage of a gold a silver should be worth, and what percentage of a silver a bronze should be worth. The applet will calculate the 'gold equivalents' for each country, and display them in their correct rank. The initial setting has been chosen to agree with the official IOC ranking system.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

[Source: Who-Won-the-2010-Olympics.ggb]

Here are some sample results:

Technical details

The above uses some of the Spreadsheet and List features of the new version of Geogebra. If you double click the applet, it should open in its own window, and from there you can open the Spreadsheet and Algebra views and explore how it was constructed.

No one, in fact … is taught at all, except so far as he is self-taught.

In an effort to kick-start this blog again, I will be picking through some material I’ve been working on recently, as well as finishing up some draft posts which have been lying dormant for far too long! First up, an interesting text on education from 150 years ago.

A while back I posted some interesting quotes gleaned from two public domain texts on education found on archive.org — The Teacher,  from 1908, and Essays on Mathematical Education from 1913. Today we step back another 50 years, to In The School Room, published in 1868, and written by John S. Hart, an American educator.

As with the other books, there is a lot of material which is still relevant today, as well as quite a lot which would no longer be applicable (such as the section about the positive results of an episode of corporal punishment). Active learning, the teacher as a guide, learning styles, and other examples of trendy ‘modern’ educational thinking all appear in the very first chapter (‘What is Teaching?‘):

In the first place, teaching is not simply telling. A class may be told a thing twenty times over, and yet not know it. Talking to a class is not necessarily teaching…

… No one, in fact, in an important sense, is taught at all, except so far as he is self-taught. The teacher may be useful, as an auxiliary, in causing this action on the part of the scholar. But the one, indispensable, vital thing in all learning, is in the scholar himself…. The teacher is to draw out the resources of the pupil. Yet even this comes short of the exact truth. The teacher must put in, as well as draw out. No process of mere pumping will draw out from a child’s mind knowledge which is not there.

… The function of the teacher is to bring about this [learning]. The means to do this are infinite in variety. They should be varied according to the wants and the character of the individual to be taught. One needs to be told a thing; he learns most readily by the ear. Another needs to use his eyes; he must see a thing, either in the book, or in nature….

Teaching, then, most truly, and in every stage of it, is a strictly co-operative process. You cannot cause any one to know, by merely pouring out stores of knowledge in his hearing, any more than you can make his body grow by spreading the contents of your market-basket at his feet…. In other words, learning, so far as the mind of the learner is concerned, is a growth; and teaching, so far as the teacher is concerned, is doing whatever is necessary to cause that growth.

The teacher who is accustomed to harangue his scholars with a continuous stream of words… is yet deceiving himself… If, after a suitable period, he will honestly examine his scholars on the subjects, on which he has himself been so productive, he will find that he has been only pouring water into a sieve.

This is certainly not the typical image you receive of the typical Victorian schoolmaster.

The paragraph which is speaking to me most at the moment comes from Chapter 17 (‘Growing‘):

The point which I wish to make, and which I deem important, is, that teachers should not rest content with their present qualifications, whatever they may be, whether large or small. Let it be the aim of every one to be a growing teacher. We come short, if we are not better teachers this year than we were last. We should aim and resolve to be better teachers next year than we are now. Our education as teachers should never be considered as finished.

If you’d like to read more of the text, then you can look at the page images on archive.org, or download a digitised HTML edition, which I have put together with the help of Distributed Proofreaders. I’ll update this post with a link to its location in Project Gutenberg, when and if it works its way through the DP system.

Downloadable content: schoolroom.html.

The new version of Geogebra has several nice new features which make it much more useful for a range of statistical uses. Firstly, it comes with a simple spreadsheet-style view, which allows you to enter and manipulate data in a grid of cells, similar to a spreadsheet. Secondly, it has a number of new statistical functions, covering a range of data creation, summation, and visualisation options.

Here is an applet which demonstrates a couple of these new features. It takes 50 random values, generated to fit a Normal(8,4) distribution, and plots a box-and-whisker plot and a histogram.

All this just uses what are now built-in features of Geogebra. My contribution is to make the histogram interactive: move the blue points on the x-axis around to alter the class boundaries. This lets you explore the ways that small changes to the class intervals can sometimes have large effects on the histogram.

Enable Java to see this Geogebra applet.

(Source: adjustable_histogram.ggb.)

Double click on a cell in the spreadsheet view to change its value. Also, just as in Excel, press F9 while in the spreadsheet view to regenerate all the random numbers. You may see the purple x-axis points move: they have been constrained to always be below the minimum and above the maximum value.

This term my school is running some maths competitions involving timed rounds. The questions are in Word, and I've been getting quite frustrated with the countdown timer built into Activstudio -- in particular, it makes resetting the count very fiddly. There are a lot of countdown/timer applications around, but the ones that do what I want seem to be either stupidly overcomplicated, not free, or to crash all the time.

To get around my frustrations I've cobbled together the following simple countdown timer, written in Javascript:

Source: countdowntimer.html.

Not particularly flashy, but it gets the job done!

You can start/stop/reset the countdown, and alter the starting count in 30s increments, just by pressing buttons, which makes it easy to use on an Interactive Whiteboard. In addition, if you click on the ^^ link (at the bottom right) the timer will open in a new window, with as few toolbars as your security settings allow.