Muzak to math by

We are in the throes of initial planning for a series of “Music and Maths” sessions aimed at 16-19 year old students, to culminate in a public performance. Using a mix of computing technologies and Blue Peter style building from scratch, the idea is to start from rediscovery of the twelve note scale and build up through construction of instruments.

The first problem we have encountered is an apparent dearth of devices or software which will listen to a note and read out its frequency. There are plenty of them (aimed at instrument tuning) which will do it the other way round, reading out a note name (C, F#, G, etc), but not a frequency. And although we did work out an alternative approach based on these guitar tuners, the interference from a building full of computing equipment, hearing aid loop generators, WiFi networks, several hundred cellphones etc, swamped them and made them useless.

A microphone attached to an oscilloscope is too unwieldy for our purpose: first introduce the oscilloscope, then explain the setting of time bases, learn to disregard noise … a one hour session would be over before anything useful had even stared. It will be useful and interesting further in, but not at the beginning.

Plan C involves auditory comparison of a tone generator signal to played keyboard and guitar string notes, by tweaking the frequency specified in the generator and deciding by consensus when a played note has been matched. This looks initially promising. We have started with NCH’s tone generator, which works well; the synthesiser at National Taiwan Normal University’s physics department also looks promising:

An alternative, offering sequential playing of different frequencies will be needed for subsequent work; a purpose made interface for preference, though it could be done using a mathematics package or even BASIC at a pinch. Ivor has written one as a Java Applet, but security measures in the browser environment where it will be used are raising barriers which have still to be resolved.

More as the idea progresses…

[contributed by Ivor McGillivray and Felix Grant]

Stonehenge - mathematics and environmental education

This is a brief description of the Stonehenge trip mentioned on May 1st this year under the heading Sun, moon and stones.

A much fuller description is provided on the Articles and papers page.

The Field Visit

A-Level and pre-GCSE Mathematics students took part in a Field Visit to Stonehenge in 1^st May 2007, one day before Full Moon. The curriculum comprised practical project-based activities integrating content from mathematics, astronomy, climate science and history¹. The party was permitted full Stone Circle Access in the evening – and an opportunity to observe moonrise and sunset from the centre of the monument. These activities were documented on film, and students were encouraged to take part in its production. The Field Visit had two main aims:

• to improve mathematics motivation;

• to afford learners a powerful affective experience of the natural world.

The latter goal features prominently in certain understandings of environmental education.

Summary of findings

• The Field Visit was highly rated by student participants.

• There is some evidence that the Field Visit improved interest in mathematics within both pre-GCSE and A-Level cohorts. In the case of the pre-GCSE cohort, however, this effect seems to have been temporary, although situational interest was stimulated on the day. This cohort seemed to especially appreciate the opportunity of using mathematical tools. Some amongst the A-Level cohort expressed a preference for contextualising mathematics within integrated project-based curricula.

• Stone Circle Access afforded a majority of student participants a powerfully affective experience. Here are some of the words that students chose to describe their experience: inspiring, fabulous, stunning, intriguing, mystical, awesome, epic, great, fascinating, indescribable.

• The experience of some individuals might be characterised in terms of cosmological based identification. For example, one student reported
  …it was like in Physics when you talk about the Universe. Inside the circle she felt small. The builders of Stonehenge were probably smaller than her. But still managed to put up those big stones. She felt small in comparison to them.
  

[1] The objective of the A-Level mathematics activity was to calculate the azimuth (bearing East of True North) of the Summer Solstice sunrise in 2000 AD, 2000 BC, 3000 BC as seen from the centre of Stonehenge using a theodolite and trigonometry. The sunrise azimuth slowly varies over millennia due to oscillation of the tilt of the earth. This oscillation is one of the three Milankovitch cycles and it is thought to have been a causal factor in the alternation of glacial and inter-glacial periods between one and three million years ago.

Beyond the Prisoner’s Dilemma

Having read Global warming and the Prisoner’s Dilemma yesterday, I spent the evening doing some fast background reading on game theory and minimax. Today I tried using the same clip with a Year 6 [10-11 years old] primary class. I, too, found that they responded well. They were animated and excited by the intellectual ideas of classification, weighting of choices, minimisation and maximisation of different outcomes. They were also interested in the general idea of using such methods to explore problem solving choices, and rapidly moved towards trying out the grid arrangement on more complex decision spaces and problems more directly related to their own experience.

One of the cases they worked on was a proposal currently under consideration and consultation for development of an area between school buildings and playing fields. Four main options have been mooted: a pair of asphalt tennis courts, a garden, or a semi wild “science area” complete with pond and simulated bog. There are also six funding options: split the available pot of money funds with a proposed new performance area in the school hall, annex all the money for the outdoor area, or work without funds and leave all the money for a better indoor development - and in each case work mount a special supplementary fundraising effort or not. So, they were planning in a twenty four cell grid like the one below.

These are my own pupils, I have known them since September, but I was astonished at how much they got from this and the degree of sophistication in their handling of it. The application to science was clearly seen and explored. Since they had followed a “funding vs benefit” example, I took them on to explore the idea of how finite public funding for science should be allocated: that, too, went extraordinarily well.

[contributed by Rose]

Global warming and the Prisoner’s Dilemma

Yesterday’s early morning email included a message from Pauline Laybourn of Minnesota, pointing me to the following video:http://www.glumbert.com/media/global

I recommend watching it through, viewing it as an educational resource. Thank you, Pauline.

Having watched the clip, I followed Mike Willcox’s ‘YouTube’ example and used it as the departure point for a discussion session with some thirteen year old students within a “Public Understanding of Science” strand.

Which side you happen to sit on the global warming debate doesn’t matter; nor does whether or not you are persuaded by the argument in this presentation. The important point is the number of themes which are here.

There is, of course, the straightforward global warming issue which the presenter is addressing. In my group of young teenagers, there was a lot of very intelligent and perceptive discussion around the examples, choices and language involved in completing the four cells of the decision grid shown on the whiteboard in the video. Are the “worst case” squares really the worst cases? Are they exaggerated? Are they understated? Are they off the track altogether? Are they both so unacceptable that the whole exercise breaks down?

There is also a very accessible entry point to game theory (game theory is a branch of mathematics, but you can go a long way in general educational terms without any explicit mathematical work). The result is an introduction to What he’s sketching out is what game theorists call a saddle point - more specifically, the type of saddle point known as a “minimax”. A minimax is a decision which minimises the maximum harmful outcomes in a given situation. A well known example of a situation where minimax may apply is the Prisoner’s Dilemma thought experiment: a good Prisoner’s Dilemma link, with an very accessible introduction leading to deeper material, can be found here at the Stanford Encyclopedia of Philosophy; other links include a Wikipedia entry, an online game at Princeton University, and a page of links connecting the dilemma to public ethics issues at the Constitution Society site.

Looking away from science to the wider context, the decision consideration process involved here is a valuable tool for thought in general. The video would be a valuable trigger for an AS level Critical Thinking session with sixteen year olds, but the critical thinking which it involves is an equally valuable component for any study, of any subject, at any school level. I plan to try it with eight year olds later in the week.

[contributed by Felix Grant]