Portable constructivism

One of my enthusiasms about ICT in education is the potential of connected systems for building genuinely constructivist activities within which learners can invent their own ad hoc subcommunities in mutual support of organised work. Which sounds very fine and impressive, and is in many ways real, but sometimes runs aground on the fact that those learners often have to leave their learning context to access the facilities for doing the constructivist thing. (I’m talking science here, but change the specific examples and everything applies just as much to arts and humanities.)

Real science.

The advantage of portable computing devices is that they encourage “real science” activities out in the world – look at X’s “Pushing up daisies” quadrat activity, for example. To have a spreadsheet available at the same time as fishing around in a ditch for tadpoles, or recording estimated speeds and accelerations of aircraft lifting from a runway, or exploring a lemonade bottling plant, brings the analysis of data vividly to life as part and parcel of the phenomena being observed. When it comes to sharing the excitement with others, though, these devices have their shortcomings.

Generally speaking, a pupil with hand held computer has to store field data in a spreadsheet or database, write notes in a word processor; return to school or home; upload both to a PC or Mac; and only then start to merge them or share them with peers.

With the trial set of Asus netbooks, I was able to take groups of students out and make the computing a seamless part of the fieldwork. There are several levels to this.

Most basic level: sneakernet.

This applies in most field contexts. Here, the pupil enters his or her own data and makes his or her own notes, as in the usual handheld setup. However, a single USB flash drive is circulated continually around the group, each pupil backing up their work to it as it reaches them and then copying a complete set of files back to their own machine. It’s necessary to name the files logically (Jesh_Kaur.doc, Jesh_Kaur.xls; John_Smith.doc, John_Smith.xls; and so on) and to avoid overwriting and keep individual work distinct, but once that habit is established it means that every member of the group has both multiple recent backups her or his own work (on both the USB drive and the computers of other members of the group) and also reference access to near current copies of everyone else’s.

The next level: WAN to go.

This was amazingly easy to set up and use, though not suitable for all settings. All that is required is a wireless router, a power supply, and a relatively small study area. When in a museum, that lemonade bottling plant, or many other visit sites, a temporary wifi zone can (with site permission) be set up in an area such as the café or visitor centre. No internet access is available, but work sharing becomes immediate. If a wifi hard disk is attached to the router, so much the better – all shared work is then available to anyone within the coverage area, regardless of whether its author is within reach. If an adaptor is carried for running the router and disk from a vehicle’s cigarette lighter, good use can also be made of time on the minibus home afterwards.

Continuity at school and at home.

If each pupil is made an author on a shared blog, with restricted readership (to avoid predation risks, but also to provide the group with privacy from nonparticipant peers) and the teacher as administrator, subsequent write ups and analysis can be pooled. By copying and pasting material from the word processor or spreadsheet such blog entries are quickly and easily generated, then can be edited and developed in place. The blog takes care of permissions – each member of a group can red everyone’s material but only change his/her own. A small portable computer continually in the same pupil’s hands, allowing work to be done when that pupil feels like it (at home or at school), able to access the blog whenever and wherever wifi access is accessible, a great incentive to participate.

Team science

All in all, my trial period with these “netbooks” has been the best opportunity yet to develop in pupils a genuine constructivist experience of working in a real community of team science. The pupils working on this pilot responded magnificently, simultaneously nourishing and feeding from each other, exchanging ideas and critiques, competing to be the best contributors to shared success.

All I have to do now is get funding to buy a full class set for long term use!

[contributed by KateQ]

Netbooks - initial hardware housekeeping issues

I have been using Psion and Palm pocket computers extensively for some years to place computer assistance in the hands of primary pupils “doing science” outside the classroom. Given a trial set of “Classmate” Asus EEE PC subnotebooks (or “netbooks”) for a month, my first concern was not their capability (obviously greater, and to be dealt with in another post) but how far they could replace their smaller equivalents in the same rôle. The two crucial issues, with small children, are portability and survivability.

Portability is a relative term. Many of the boys to whom I loan a palmtop machine simply put it in their trouser pocket. Girls, on the other hand, usually put it in a school bag along with their books and so on. These Asus machines are about twice the size of a Psion, four times that of a Palm device. That makes them unpocketable, but doesn’t much affect a school bag. For boys, then, a change in behaviour is often necessary for these machines to be considered “portable”, but not for most girls.

For that reason, I loaned all five machines out to boys on 24 hour tickets in the first week just to see what would happen. In most cases, they went into sports bags (and came bag muddy) or satchels (and came back covered in grey fluff). In a significant minority (15%) of cases they were carried around continually in the hand, which places them at considerably greater risk (but see below on survivability).

After the first week they were loaned as required, regardless of gender or time span; as expected, the girls treated them exactly as if they were palmtops.

Survivability was more worrying, and I asked how much risk was acceptable in field trialling. The answer back from the sponsor was that deliberate attempts to test a machine to destruction would be unacceptable, but that we shouldn’t let potential hazard stop us from doing things we would do with a palmtop. It happened that a joint maths/sport project was under way, so the trial subnotebooks were added to the stock of Psions and Palms and allowed to go out onto football and netball pitches.

A football pitch provided the severest test of survivability. A pupil took one of the netbooks down to a practice game to try out both real time analysis of game descriptors entered into a spreadsheet (OpenOffice Calc, saving in Excel file format) and video capture to disk using the built in camera. The computer’s novelty attracted a lot of attention and it wasn’t long before attempts were made to take it away from its guardian, who resisted. In the resulting mêlée the computer was dropped, trampled on by several sets of studded boots and rolled over by half a dozen tussling nine year old boys. When order had been restored, the referee had to dig it out of the mud. Cleaning the mud out of USB ports, Ethernet socket, VGA output connector, sound jacks and, worst of all, the keyboard, took a lot of time, patience and cocktail sticks but, miraculously, everything was still in perfect working order. After that, we sealed all orifices with electrical insulating tape unless they were needed for use; proper sealing plugs would be better still, but would probably get lost fairly quickly.

Fast forward: despite horror stories like this, and my gut feeling that these machines are not ultimately as robust as handhelds, none came to grief in the time we had them.

My summary judgement: these are a valuable addition to the portable computing options available for primary science. Since Psion type machines are no longer made, and can only be replaced second hand, their gradual replacement on failure by these small subnotebooks seems a good strategy. At the same time, it would be a mistake to withdraw a working handheld (especially of the palm type). For as long as possible, keep existing palmtop hardware in use but expand enthusiastically with subnotebooks.

[Contributed by Chandra]

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.

InspireDaisies

I have a standard data collection activity, borrowed from AbsentCat, which I call “Pushing up the daisies”. That’s not a very good name, bearing no relation to what actually happens, but it has the virtue of amusing pupils.It’s a quadrat exercise. Each pupil takes a pen, an old sock rolled into a ball, and a sheet of A4 card with a 100mm square hole in the centre of it. We all go to the centre of a convenient expanse of grass, form a circle facing outward, and throw our socks. Where the sock lands, put your sheet of card and count how many daisies are visible through the hole. Write the number down on the sheet of card, throw your sock again. Repeat until the novelty wears off, then return to the centre of the grass area to collate the results.

Sometimes, with a small group, I will replace both card and sock with a frisbee in the centre of which a circular 113mm hole (to match the area of the 100mm square) has been cut.Throwing things around in the open air is always preferable, on a sunny day, to being indoors. We usually take a picnic along, and a set of palmtop computers, so we can conduct the subsequent analysis of our daisy data in relaxation amongst the daisies themselves. This approach pays dividends: I get a lot of good natured work out of children who would get bored and impatient if we did academically equivalent work indoors.

This week, instead of the palmtops, my year fours (age 8-9) took a laptop with InspireData (reviewed here). Instead of writing their results on the card, and collating them later in a spreadsheet, the pupils brought each count back to the laptop and typed it into InspireData’s data entry “questionnaire”. Each observation was identified by the child’s name, and a photograph of a daisy was imported to replace the standard marker, so as the session proceeded we watched a growing histogram of labeled daisies gradually assemble on screen.

The class kept on gathering data much longer than usual, keen to see their name on screen as often as possible. Result: a much larger results database than usual, and more pupil involvement in the analysis phase.

I plan to follow up, at the end of this week, with botany and geography lessons based on the results using the InspireData histogram as a reference point for analogy with quantitative methods in both of those fields.

“Pushing up the daisies” is a good educational activity, offering a number of painless entry points to maths and science topics. InspireData adds immeasurably to it.

[contributed by Sayid]

Polaris and me

I was going to review Polaris, a science fiction novel by Jack McDevitt. I’ve also been asked to write about what has happened to me since I reviewed Sunstorm as well. They have a lot to do with each other and I don’t think I can do them separately. So am doing them both together, and I hope it makes sense.

Before my English teacher recommended Sunstorm I was not interested in maths or science at all. In this essay I am going to save a lot of explanation by just using bold type to show things and ideas which are new to me since I started reading Sunstorm. I am glad that I was told to use a pen name, because if my friends knew I was writing this I would be socially dead forever.

After I reviewed Sunstorm, I read Donna’s review of Seeker. The thing that I liked most about Sunstorm was the idea of a planet being fired across space to hit a sun, like a stone being fired at a target with a catapult. Then my maths teacher showed me how to model this on a computer, and I realised that it’s actually more like firing the stone from a catapult in London and hitting a melon in Australia or somewhere. Anyway, Donna’s review mentioned that something similar happened in Seeker, so I read that as well.

I found that Seeker is the last book in a set of three about the same characters (the first is A Talent for War and Polaris is in the middle). So then I read the other two as well. All of the books have the same pattern: there is a mystery, the main characters discover it through something to do with the antiques trade, historical research gets them close to solving the mystery, and the mathematics of moving bodies finally gives them the answer. The mysteries are all different, and make you want to read to the end, but I won’t spoil them by describing them here - and anyway, it’s the maths bits that interest me (I never thought that I would hear myself say that). The historical research interests me too.

In Seeker the maths was about how a stellar system is affected by a brown dwarf star passing close by. In A Talent for War, it’s where a spaceship would be after two hundred years. And in Polaris it’s sort of like a cross between Sunstorm and Seeker because a small but super dense star called a white dwarf hits an ordinary G class star like our sun (not deliberately, it just happens) and goes straight through it and out the other side and destroys it.

I have got totally into this moving bodies stuff. I find the ideas exciting. My maths teacher has shown me how to find information about it and I have done a lot of reading. He has also shown me how to use a spreadsheet and a program called Autograph to set up and investigate my own models. I have learnt a learnt a lot but the the biggest thing I’ve learnt is that I have gone as far as I can without learning some pretty scary maths.

I have started studying some AS maths modules on my own. Well not really on my own because my maths teacher is helping me before school and my uncle is helping me at home but I mean not in a class or anything. I have completed module M1, which is the first mechanics module, and started on M2. Mechanics is what they call the sort of maths that will eventually let me cover orbits and trajectories and stuff (M1 and M2 don’t get that far, but I need to understand the basics). To understand some of the mechanics I need other maths, called pure maths, which doesn’t have anything necessarily to do with mechanics but you use it as a sort of way to describe things - my English teacher pointed out that it’s like I can only enjoy poetry if I can already read. So I’ve done quite a bit of P1 as well (that’s the first pure maths module).

I am using some software called Derive to help me with understanding the maths I am doing. There’s a lot of other software as well and none of it would be so exciting without the models which they let you build to try things out.

I’ve done a little bit of calculus with my maths teacher and my uncle. Calculus is when you imagine very small bits of a problem so you can get your head round it, then imagine that small bit happening over and over again, forever, to make it back into the big problem again but now you understand it. I haven’t explained that very well, but it’s important and it works. Its how you can start with the velocity of something, and the gravity of a star pulling it, and see where it will go, or the other way round.

By September I think I will have finished all three AS modules. My uncle says I could take the AS exam, even though I won’t have done my GCSE yet. But that would totally blow my cover and everyone would think I was a geek. My teacher says he’ll see if I can take it somewhere else that nobody knows me. I don’t know. I’ll see.

Doing all this other stuff has made me better in ordinary school maths and science too. I used to be rubbish at algebra, but now it seems easy. I know now that when you do experiments you do them lots of times and then look at all the results, not just one, and now the handling data part of maths makes sense too (but I don’t want to do the S1 statistics module cos that looks really scary).

My maths teacher has set up some experiments for me, like rolling a marble across a rubber sheet on a frame. You can poke your finger into the rubber, or put a lead weight on it, and pretend the dent is a gravity well and see what happens when the marble (which is supposed to be a lump of rock in space) passes near it at different speeds. And we tried firing an air gun through an egg in front of a video camera to see what might happen when the white dwarf goes through the G type star in Polaris, which is a physical model instead of the mathematical models which you do with pen and paper or with software.

I’ve started to think about what I want to do in my life. I am still most interested in literature and drama but I’m interested in other things too. I’ve been doing paintings and models from the shapes that all the trajectory models make, and imagined using them for stage sets - weird or what? I just tell my friends they’re abstracts. Because of these novels by Jack McDevitt I’ve got really into history as well, and I’ve seen the same sort of graph shapes in history books as in mechanics, like the way population grows looks like the way a rocket’s height changes as it takes off.

It would be nice to do everything, but I’m not sure you can. People seem to do one thing or the other. Mr Grant who organises this site and asked me to write about this stuff says he did literature as well as maths and sciences when he did his A levels but he’s quite old and I think things have changed since his day. He says that people who write books like Sunstorm and Seeker need to understand the maths and science as well as being able to write, and Jack McDevitt must understand history too, and I suppose that’s true. But A levels are a long way yet. I don’t even start my GCSE subjects until September.

Well, that’s a little bit about Polaris and quite a lot about what’s happened to me since I read Sunstorm. I hope it wasn’t too boring. And I hope nobody I know ever realises who I am.

[contributed by Lakshmi]

McDevitt, J., A talent for war. 1989, Sphere. 0747403333.
McDevitt, J., Polaris. 2004, New York, Ace Books. 0441012027.
McDevitt, J., Seeker. 2005, New York, Ace Books. 0441013295.
Clarke, A.C. and Baxter, S. Sunstorm: A time odyssey. 2006, London, Gollancz. 0575078014

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]

Beanbag Thrower still mid-flight

Sorry, everyone: I had hoped to have the Mathematica 6 Beanbag Thrower packaged and submitted to the Wolfram Demonstrations Project this week, but time has run out on me. I shall do it as soon as I can. It’s the packaging to Wolfram’s specification that I haven’t yet come to grips with - I have had offers of help, but want to get it done myself. Watch this space…

[contributed by Chandra]

Throwing beanbags in Mathematica 6

This is the first in a series, illustrating how Mathematica 6 makes it possible to package complex ideas in a way which is suitable for primary education use. You do not need any mathematical knowledge to use the ideas and materials here. Nor do you need any knowledge of the Mathematica 6 program, although you will need a copy of the software.The equation for motion of a projectile on the horizontal plane is not normally encountered until level three in either mathematics or physics. The intuitive idea of parabolic motion dependent on velocity and angle of launch, however, is easily grasped much earlier. In this case. we experimented with class of 8/9 year olds in a British school, as part of an integrated sports theme.

We started by having one pupil toss a beanbag to another, from one side of a classroom whiteboard to the other, while the rest of the class watched and commented. The shape of the path followed was sketched by each child.

This was then repeated several times, the teacher marking one point on the trajectory each time until the shape emerged. The class discussed the similarities and difference between the shape on the board and their own sketches.

Next we moved outside to the playground, where sheets of flipchart paper had been taped to the side of a school building, from ground level to a height of 2 metres, over a distance of five metres. The beanbag throwing was repeated, but this time the children did the marking in of trajectory sample points themselves. In groups of five or six, they spaced themselves between thrower and catcher and each marked the point at which the beanbag passed in front of her or him; after each throw, a new group replaced the last until each pupil had made one mark. Once again, we discussed the shape of the curve.

The throwers were now asked to throw the beanbag higher, but still to try and drop it into the catcher’s hands. The changing shape of the parabola was watched, and discussed, by teacher and class.

The catcher was then removed, and each child took a turn at throwing. They were encourage to vary the strength of their throw, but to aim always just above head height. The effect of throw strength on range was one they already understood, but the changing yet conserved shape of the trajectory as new to them.

Having established the idea of curved trajectory, affected by both angle and strength of throw, the action moved back indoors where the pupils worked in threes and fours. In the centre of the room was laptop with a copy of Mathematica 6, showing a graphic demonstrator (prepared beforehand) which was projected onto an electronic whiteboard.

The demonstrator, which we had called The Beanbag Thrower, allowed the pupils to experiment very rapidly with different combinations of strength and angle of throw, by adjusting “slider” controls. (Double click the illustration on the right to see it full size.)

We decided not to issue specific prompt material at first (though we had them ready, just in case), preferring to see what would spontaneously emerge from free experimentation. WE gave each child a sheet with two columns labeled “what did you do?” and “what happened”, then left them to devise their own strategies. This optimistic but risky approach was vindicated when, after less than minute, the first group announced that “setting the angle half way always sends the beanbag furthest, whatever strength you throw at”. The rest of the class immediately switched to testing this hypothesis, and quickly confirmed it.

The next surprise was when another group correctly interpreted the meaning of negative y values, where the curve drops below the start point. “That’s where nobody catches the beanbag” said one girl, “and so it falls lower than your hand was when you threw it”. Her friend extended this to “And then it goes on falling, lower than your feet, like if the ground wasn’t there”, to which a boy added “Like if you were on top of a building, and your throw took it over the edge”.

Clicking a “+” symbol at the end of a slider opens an input box and a small set of CD player type controls. The box lets you type in an exact number for height or angle, so the pupils were able to check that maximum throw distance really did occur at exactly half way along the angle range, not just approximately.

The CD type controls bar lets you run variable through its full range automatically in various ways. When they discovered this. the pupils’ reaction was to set both height and angle running simultaneously, as fast as possible, in opposite directions - which, while fun, didn’t reveal very much. After a while, though, they realised the value of setting one variable to a fixed value and letting the other roam repeatedly up and down its range in slow motion. Watching the height, the range, and the value of the changing variable, they came to several useful insights. When they realised that you don’t have to maximise the range but can find an infinite number of ways to deliver the beanbag to a particular point, they became very excited: this related directly to football, netball, cricket, tennis, badminton, and darts.

They were also intrigued by the realisation that, for any given distance and strength of throw, there are two possible trajectories. There was much animated discussion over the relative advantages and drawbacks of high versus flat trajectories, eventually leading to understanding that the flat option usually give a competitive advantage.

We, too, were excited by the outcomes of the experiment. The class teacher felt that her pupils had learned lessons in sports, ICT and applicability of mathematics, not to mention transferable skills in group work, co-operation and the efficient conduct of a scientific enquiry. I felt that I had a new range of useful possibilities to explore in wedding maths and science to wider curriculum through IT.

You can download our Mathematica demonstrator model here (see bottom of this post), to experiment with it yourself in a copy of Mathematica. Unfortunately it will not run in the free Mathematica Player, as such graphics are only usable in a special file format can only be produced by the publishers, Wolfram Research . As soon as I have time (during the Spring break in a couple of weeks, perhaps) I hope to work through the process of submitting it for publication as a demonstration for the player.

There are, if you want to try them out, already some projectile models on the Wolfram Demonstrations Project site, but they are more complicated and not (in my view) suitable for primary school use. The exception is “Dart Practice”, which I plan to try with older (age 10/12) pupils.

• The free Mathematica Player can be downloaded from http://www.wolfram.com/products/player/
• The Dart Practice model (usable in the free player) is at http://demonstrations.wolfram.com/DartPractice/
• The Wolfram Demonstrations Project site is at http://demonstrations.wolfram.com/
• You can look at all of the available projectile models here.

You may also like to look at the Kinetic Books Virtual Experiments which were reviewed by AbsentCat last month.

The record sheet and Mathmatica notebook which we used are in a zip file which you can find here. If you are interested in the works behind the demonstrator, and brave enough to tackle the level three mathematics involved, there is also full explanation sheet in the same zip file. If you are unable to use the zip file, send an email to the editorial address for these education pages (you can find in on the “contributors” tab) with “Beanbag materials request” in the subject line.

[contributed by Chandra]

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]