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.

Sun, moon and stones

“Can we go on a trip?”

It’s the perennial cry of every class, and educationally it pays dividends, but mathematics students tend to lose out to more field based subjects like Biology, Geography, Environmental Studies and English literature.

Where can a maths class go, that is both useful and enjoyable?

Today I spent the afternoon and evening with a mathematics lecturer Ivor McGillivray, a film team, a group of students studying maths at GCSE level and A-level, two laptops and an electronic theodolite, at Stonehenge - the megalithic ruin on Salisbury Plain in south western England. The focus was the history of sun and moon in mathematics, and practical activity was central.

It was an eye opening trip, inspirational and remarkably successful in pedagogic terms.

It’s not my story, so I won’t go into detail now, but I hope to persuade Ivor to write an article for us.

Photographs at left (click them for a larger view) show, from top…

• Calculating the height of a standing stone.

• Cross checking measurements for accuracy.

• Setting up the theodolite.

• Relaxing between tasks.

[Contributed by Felix Grant]

Virtual experiments from Kinetic Books

Supplier: Kinetic Books, http://www.kineticbooks.com.

One of the challenges in tackling the declining popularity of science subjects throughout education, or seeking to increase the scientific literacy of those who will not be scientists, is how to make experimental science concepts accessible, fun and relevant. Tapping into the skills and environments which young people already inhabit is one very good way to tackle that challenge.

Kinetic Books offer a system of online or CD based textbooks and virtual labs; I was particularly interested in the Virtual Labs, and concentrated mainly on those. The system is explicitly designed for learning across a range of physics topics, but the way they are presented makes it very easy to incorporate selections from the material into other courses too. Mathematics, of course, is an obvious beneficiary, but scientific thinking components can be introduced or strengthened within other areas from social studies through critical thinking and public understanding of science to art history.

There is a core of instructional material, with good use of hypertext sidebars offering expanded information plus frequent check and stimulus questions. There are also links to material elsewhere, and graphically simulated experiments. It could be used as a self study resource pure and simple; there will be contexts in which that is appropriate, but for me the strength lies in the ease with which bite sized parts can be used to enrich other approaches.

The levels of mathematics involved encourage this second view. Learners do not need calculus, but are expected to be comfortable and fluent in manipulation of inverse quadratics. The interactive simulations, on the other hand, could be used alone to develop intuitive understanding at any level from infant school upward. Selecting portions in this way, I’ve experimented successfully with learners aged from 8 to 34. There is also the question of national differences in curriculum; British teachers would find frequent discontinuities between US and UK content if they tried to work exactly to KB’s structure without adaption.

For me, the simulations are the real centre. Using graphics to good effect they provide the opportunity for hands on experiment with a range of models which are difficult or impossible to set up physically, and hard to observe reliably.

The motion of a simple projectile can be modelled easily enough using a bouncing ball, but monitoring the velocity and position of that ball with any precision requires either video recording or specialised equipment and lots of time. Getting access to a helicopter is usually both difficult and expensive. Orbital mechanics are entirely beyond any realistic classroom or lecture theatre environment. Using Kinetic Books’ virtual physics lab, all three become very quick and trivially easy to explore, with unlimited reruns allowing deep exploration in the time needed just to set up a ball bouncing experiment.

The simple projectile is modelled as a cannon ball (one dimensional motion having already been covered beforehand). First it rolls out of the muzzle and falls vertically to ground. Then, by adjusting the muzzle velocity, the learner attempts to drop it into a pile of sand some distance away - unsuccessful attempts remaining on the ground where they land, as markers, while trial and error brings subsequent shots closer and closer until the sand pile is scattered by a direct hit.

The cannon starts in a fairytale Arthurian style castle, then later appears on a globe as Newton’s Cannon for the first introduction to orbital and escape velocities. After that, it is replaced by the moon - which, in a game style setup, must be restored to orbital velocity before it falls and destroys the Earth. Further simulations involve docking of two spacecraft on different orbits, the twin moons of Mars, and so on. The orbits concerned are not simple geocentric circles, either - Deimos, for instance, changes its elliptical motion in relation to both Mars and Phobos, its velocity visibly changing between perigee and apogee.

I’ve concentrated on projectile motion because it is a key part of the freely available trial material, but there are plenty of other topics - waves, thermodynamics, electricity and magnetism, light and optics - at levels from the concept of measurement to special relativity and quantum or nuclear physics.

Pricing is realistic in comparison to other resources, and can be managed in various ways to suit different usages - even light use will justify the expenditure on perpetual licences, and individual private copies are affordable by any student who already buys course books. The experiments rely on Java, Quicktime and Flash, but those are free downloads. I hit an initial problem with some of them not displaying correctly, but response from Kinetic Books to my call for help was prompt and effective - the solution is a simple tick box in Quicktime’s setup.

Nothing in this world is ever perfect, and a review wouldn’t be complete without mentioning a couple of minor reservations, and the textbook entry on SI units illustrates both.

The importance of “powers of ten” is presented, and 1000 metres in a kilometre is given as an example (though this is an American text, so be prepared for US spellings of “meters” and “kilometers”). The principle of ten to the power three as a standard spacing, however, is not made clear without following further links.

Then there is the embedding within a wider, nonscience cultural context. This is one of the things I really like about Kinetic Books, and a reason why I would recommend them, but it has its tightropes and pitfalls. For instance, while I am very glad to see the origins of the SI set in the larger picture of revolutionary France, I might have preferred students to decide for themselves, rather than be told, that the “revolutionaries were a little extreme (as revolutionaries tend to be)”.

But, I repeat, these are minor details in a well designed and thought out whole which I recommend.

I’m very grateful to Donna (see contributors page) for pointing me towards these resources.

Supplier: Kinetic Books, http://www.kineticbooks.com.

[Contributed by AbsentCat]