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]

Testing equation editor responses - results

Having marked the physics assignments submitted during my mini experiment (see Testing equation editor responses), after some delay caused by the flu which is doing the rounds, I sat down to look at what they revealed. Questionnaires were given to the students after hand in, disguised to appear as enquiry into attitudes and responses to aspects of school itself rather than the equation editors, supplied some valuable information about students viewpoints and inclinations. Information form other staff, including assessments and reports, provided a third reference point.

Taking all of that together, the results broadly corresponded with Lakshmi’s perception.

Students whose favourite subjects include the visual and dramatic arts, and whose best marks are in those subjects, tended to handle Equations! with more confidence than MathType, and to produce better designed physics assignment pages when working on the machine on which it was installed. Interestingly, this was also true of those whose focus is physical activity (games, sports, physical education).

Students with a preference and bias towards English Language, literature, history, geography, and sociology showed the reverse inclination: they performed best, and felt greatest confidence, when using MathType.

Surprisingly, the split was also visible within the subgroup of students who prefer and perform best in the sciences. Students whose chemistry is stronger than their biology had a MathType leaning, while their peers who lean towards biology but have a weakness in chemistry preferred Equations!. Those whose strength is in physics and/or maths, however, were indifferent to which package they used, were equally competent and confident in either, but showed irritation at having to switch from one to another, in either direction, when resuming an assignment on a different machine.

One final split emerged. Formulator Express is freely available to all students on all other school computers apart from the two laptops which they were required to use for this assignment. In roughly equal numbers, some students preferred either of the trial packages to that established option while others reacted against the need to shift away from it. None of them placed preference for their usual tool above one of the trial packages but below the other - either they preferred it to both, or they didn’t.

[contributed by Ross]

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]

The joy of equations

Part of my summer holiday was spent in trying to learn something about stuff outside the textbook areas of maths I’ve been looking at. They are fascinating, but because I’m still an arts and humanities girl at heart I needed something more romantic to lighten them up a bit.

My history teacher showed me some examples of how models can be used to try out ideas and see whether they fit what really happened in the past - for instance, I’ve played with a set of equations for the expansion of the Mongol empire mentioned in Sunstorm, and the spread of the Black Death in fourteenth century CE Europe. He also introduced me to sociology, where equations describe the behaviour of large numbers of people.Anyway, to get back to scientific computing, I find the way equations are written very beautiful but the way they go into a lot of software programs is ugly (especially spreadsheets). I often need to write them out myself before I can relate to them. Mr Grant lent me a computer with several programs which just write equations, the way you would by hand but typing them on screen. I’ve also been given a school copy of a free one (supplied by the government education ministry) to use on my own computer. I’ve had a lot of fun with these programs, and they have made the final connection between the excitement I feel about physics models and the “aesthetic me” that loves poetry and drama and painting.

The free program is Formulator Express, and is part of a set of programs given to teachers. I am very glad to have it for my own, but I hope to get my own copy of either MathType or Equations! (both of them have to be bought, but my uncle is talking about getting me one for my birthday). They are both very good, and do more than the free program, but I think different people would buy them. MathType appeals to the part of me which likes to write words, and Equations! pleases the bit of me that likes pictures - equations are both descriptions and pictures of something I can’t see with my eyes, only in my head.

All of these programs come down to picking and combining symbols, then letting the computer take care of drawing, spacing, arrangement and so on. The result is wonderfully sensual, with all the curves of a proper font setting off the beauty of the equation itself. They give you all sorts of ways to control and fine tune the way the equation looks, but won’t let you break the rules which control how an equation is supposed to look. They are magic. They have all the best bits of hand writing equations but let you adjust everything until it’s just right.

Equations! and Mathtype both help you to do a techie language called LaTex as well. I don’t think Formulator does, or if it does then I haven’t found it. I’m only just starting to figure this out, but it’s a way to describe equations. I’ve sort of got my head round the basic idea, but I don’t think I’d ever have the patience to get good at it - so it’s a good thing that these equation editors do a lot of it for you. For myself I found Equations! best for this part, it seemed more like the way I think, although MathType does whole pages of stuff at once.

I said before the summer that I had started painting equations. The equation editors have encouraged me to develop that work, and I have several sketchpads filled with arrangements of equations and graphs combined on the same page. (Apart from being beautiful, this is also useful. I tried to make sense of Einstein’s relativity stuff from a book, and got closest to understanding it through my montaged watercolour sketches.)

Now my English teacher (who started me on this stuff in the first place) and the art teacher have suggested that I work up some of my sketches into background scenery for a German play called Die Physiker, about Newton and Einstein. I am worried about this, as I don’t want to get known as a geek, but the idea does make me feel excited. I have done some experiments in the drama studio after school this term, putting Equations! equations and Autograph curves from the computer onto large sheets of calico to see how the forms and dynamics work together on a large scale.

[contributed by Lakshmi]

• Autograph and Equations! were supplied by Chartwell Yorke (who also stock MathType).
• Formulator (from Hermitech Laboratory in the Ukraine) is licensed for educational use as part of a standards pack from the UK Department for Education and Skills. The free version used by Lakshmi, Formulator Express, can be downloaded, or a full version purchased.
• MathType was supplied by Design Science.

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]

InspireData (review)

Inspiration, the mind mapping software, is widely used in education. InspireData is a new addition, in this academic year, from the same publisher.The principle behind InspireData is much the same as its established sibling: visual learning by direct manipulation through an intuitive interface. I’ve never seen anything to compare with it: data are entered (or copied and pasted) into a conventional looking worksheet, instantly familiar to an Excel user, but nothing after that resembles what you may be used to in a spreadsheet, graphics program, or other data manipulation package. In trials with pupils and students aged from eight to eighty three, over the past few weeks, I’ve found it uniquely effective.

When you first switch from the worksheet to visualisation, you will find your data points scattered randomly all over the desktop. I found that this works well with introductory sorting exercises with found objects or record cards - especially if you start by applying a Venn diagram.

I say “applying” a Venn diagram, not “drawing” one, deliberately. Everything you (or the student) do here assembles itself before your eyes, each data point moving across the screen from its random initial position to the appropriate place in the graphic. Click the on screen Venn diagram button twice, to create two set loops; click each loop in turn and define them as “male” or “female”. Assuming that you have entered the name and gender of each pupil as your data, the points will travel quickly (but not too quickly) across the screen and cluster in the appropriate loop segments. Now switch on data point labels with another click, choosing “name”, and each point will show which pupil it represents. Now each member of the class can watch her or his own personal avatar move about in subsequent work.

Now click the stack diagram button. The Venn loops disappear, the points move again, and when everything comes to rest your pupils are stacked up in two bars above “male” and “female” markers, graphically showing the gender balance of the class.

Everything works the same way. If you entered the heights of your class members in centimetres, along with their genders, click the variable used for that stack chart and select “height”. More visual rearrangement, as the names shift around to align with the height bands which appear across the x-axis to replace the gender labels, for a schematic histogram. Select colouring, and the point beside each name changes hue to reflect gender - blue for girls, red for boys, perhaps. The way height is distributed by gender is immediately there for discussion. You can, if you wish, take the colouring back into a Venn diagram but this time define the loops as (for example) “height more than 120cm” and “height less than 150cm”, then discuss the way genders divide across the three set segments.

Pie charts work the same way. Leave the gender colouring in place, and define the sectors of the pie to reflect height bands - maybe start with the same three, then add more to increase the resolution as discussion develops. With each change, the names will shuffle about the screen to adopt their correct positions.

This needn’t seem to have anything to do with maths, so it’s a wonderful way to painlessly develop categorisation and quantitative vision alongside science as fun - possibly in an apparently nonscience context. I spent a session with a ten year old soccer team, feeding in their own choice of vital statistics for their personal heroes (club, field position, age, height, weight, number of goals last season…; for Beckham, Gerard, Rooney…).

Though I didn’t use it here, there is the facility to use custom icons (either across a whole variable or case by case), so a small photograph of each player would have been a valuable addition. Discussing the patterns which InspireData threw up, they generated their own questions, hypotheses, lines of enquiry. One of them had read a rule of thumb for ideal relation of height to weight - and InspireData moved the players (colour coded by performance) into a scattergram. Then, two hours in, one lad said: “could we use this for maths?”Getting the information into the worksheet is simplicity itself. There is a simple data entry form, called “Questionnaire”, into which each student can individually type their chosen information without having to navigate the worksheet at all. You can, if you wish, add helpful comments to each field (such as “how many goals did your player score last season?”). The user types into clearly laid out boxes, edits until they are happy, then a click commits the result to a row in the sheet.

For its purpose, and its level, I can’t praise this program highly enough. If you do any kind of data handling, in any subject, at any level where your learners are new to data analysis and would benefit from a visual approach, buy it.

[contributed by Felix Grant]

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]

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]