Saturday, October 21, 2006

More on blackboards

(This post was on Bill's stuff but I have moved it as being more related to science and education)

A conversation on the train today on the subject of black boards. My friend Curtis asked how the new Maths building was going and if we had our blackboards sorted out. As we have been having a lot of blackboard related discussion prompted by Sasha Borovik's blog "Psychophysiology of blackboards" it led me to think about similar cases where a high technology alternative is proposed to something very simple and effective, and while the high-tech solution has many supposed advantages it simply does not do the basic function as well as the original.

An interesting example is the case of "electronic books". The idea is to make a small portable device that can store images of (perhaps several) books electronically and display the pages on a liquid crystal display. Such things exist, and they have advantages including: not needing to cut down trees (one would have to do some careful accounting to see if there was a net environmental benefit), the possibility of taking lots of reading material while travelling without the weight, the possibility of rapid electronic distribution over the Internet. However several technological challenges mean that that they do not yet work as well as a paper book. A recent contender is the Sony Reader. The resolution of portable screens is no-where near that achieved in printed media. Interestingly I was involved in a research project with HP Labs Bristol on liquid crystal displays. One aim is to make a bistable liquid crystal display so that pixels can be turned black or white without the need for to be frequently refresed. There is a bit about the project here. For "electronic paper" this is essential to provide the required resolution, and the page only needs to be changed at the rate of turning a page so slow is ok. But it is not yet "in the shops". Also contrast is a problem. Most laptop computer displays require a back light, and cannot be read in strong sunlight and this too is a problem for electronic books.

While we might see these problems overcome in the near future it currently the high technology replacement simply does not provide the basic functionality required of a paper book. And so it is with the humble blackboard. There are some very interesting "smart boards" widely deployed in schools. This consists of a touch sensitive screen on which an image is projected. A computer stores the lines drawn on the screen with a special pen and projects them on the screen. They have the advantage that previous screen-fulls of writing and drawing can be replayed, very much like the traditional over-head projector scroll. As the image is stored electronically it can also be made available to the students. I have not seen these devices deployed on the scale needed for a 200 seat university lecture theatre in which the entire wall is covered by several moving boards. Although technically it is feasible. However as a basic medium for writing it still does not approach the functionality of a blackboard. For example to writer must be careful not to obscure the projection, otherwise she cannot see what she is writing. Also the lights must be dimmed (or sunlight blocked) for the screen to be easily seen. This makes it harder for the students to take notes and increases the tenancy for them to fall asleep.

I can see perhaps that in a few years with as substantial effort, it might be possible to make an electronic system that achieves the functionality of a blackboard (as well as also having some advantages). But the result would be likely to be quite costly, require more maintenance, and provide only a marginal advantage over the low technology approach.

As an afterthought I wondered about electronic pianos. Not such a clear case as for piano players, and listeners to pianos up to a certain level a modern electronic piano provides an acceptable alternative in terms of its basic functionality. And of course has some additional functionality such as staying in tune, recording keystrokes, playing an accompaniment , etc. Interestingly electronic pianos are cheaper the conventional pianos again making the comparison with blackboards and books not so relevant.

p.s. 10/10/2006 Sasha Borovik's blog on blackboards gets mentioned in The Guardian

Thursday, October 12, 2006

How many "centres" are there of a triangle

One of my students wrote the centre of a triangle is (r1 + r2 + r3)/3. Well that is one definition, the centroid or centre of mass if it is has uniform density.

I immediately thought also of the

  • Circumcentre, the centre of the circumscribing circle

  • Incenter, the centre of the inscribing circle

  • Centroid, where the lines joining the mid point of each side to the opposite vertex meet.


Then some googling found me Jim Loy's page listing a few others


  • Orthocenter where the three altitudes of a triangle meet.

  • Gergonne point where the lines connecting the tangent points of the incircle to the opposite vertices meet

  • Fermat point (also called the isogonic center or the Rorricelli point) which minimizes the sum of the distances from the three vertices.



and three more as well! Then I found an even better page Geometry step by step from the land of the Incas(!) Listing 16 different kinds of centres of a triangle (with pretty colour pictures).