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]

Testing equation editor responses

Following Lakshmi’s post on use of three equation editors, and a conversation with AbsentCat about discussion with the author of Equations, I was curious about how differences in formatting assumptions are perceived by users.

Equations implicitly assumes that the host application (word processor, web editor, or whatever) will see to arrangement of completed equations in relation to its own design priorities. MathType, on the other hand, assumes that a given equation system will be arranged according to a chosen set of mathematical conventions, indepenent of the context within which it is to be placed.Both assumptions have arguments in their favour; but they are nevertheless distinct. I wonder whether there is a possible link between them and Lakshmi’s observation that MathType appealed to her verbal side, Equations to her visual sense.

Yesterday I started a small experiment. I am trying out both programs on a class of fifteen year olds typing up a short physics investigation. None of them has used an equation editor before, so they were all given a training session on both products. They have now been told to use one of two otherwise identical laptops, always available in the lab, to type up their work in booked sessions over the next two weeks. One machine has Equations and the other MathType, but neither reveals which until after log in, and I shall randomly switch their positions. I hope that subsequent questionnaires will show what (if any) differences emerge in their responses.

[contributed by Ross]

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.

Cabri3D: building big models on small beginnings

Over the summer, I spent a lot of time getting to know Cabri3D better, after the success with a simple net demonstration.

Truancy work has to continue through holidays - not at the same level as term time, perhaps, but there must be some continuity or the youngsters disappear you simply lose all that you’ve done. So, there have been drop ins and workshops at intervals over the summer. I used some of this time to get my young clients exploring Cabri on my behalf, letting them teach me - something which engages them in a way that a lesson the other way around can rarely do.

They particularly liked the “models” class of packaged examples, and that led to a lot of impromptu work in which I hastily learned about some of the ideas embraced by Lakshmi in earlier posts. They were fascinated by the basketball example, in which a single bounce through the hoop is repeated and rotated through 360 degrees. They also made the link for themselves between this sort of mathematical modelling and the animation of computer games - in fact they commented, without my prompting, that movement in video games is generally less realistic than the Cabri3D bounce or “Claude on a swing” and “Claude on a Trampoline” which cracked them up. The GPS system model appealed to the boys (though not the girls) as a techie toy.

Several of the girls were fascinated by “Escher’s stairs”, and that was their way into the actual works of Cabri3D - they wanted to know how it was done, and set about finding out. The boys were then challenged by macho pride into exploring how to do it as well. So now all of them are conversant with the Cabri3D innards, and are making progress with teaching me. Models have also, as a result, become a regular talking point, and basic maths is improving visibly in consequence.

All of which I call a worthwhile result

[contributed by BobTheBumbler]

• Cabr3D was supplied by Chartwell Yorke

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.

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

Mathemagica - Mathematica Player completes the magic square

I have, in the past, seen the effective use by contributor AbsentCat of magic squares in a remarkable spread of contexts. From the moment they learn to add three single digit numbers together for a two digit answer (the row/column/diagonal sum of a 3×3 magic square is 15), children are fascinated. The intellectual appeal can still be triggered at any age above that - I have seen it enthuse a mixed truancy group with ages from 10-16, a hospital education group containing a very sick 18 year old cancer patient, and a pensioners’ Third Age study group. Only the management and presentation needs to change.

And the magic square is not just an entry point to mathematics: it has ramifications for almost every other curriculum (and wider) context.

Having seen this success I have, naturally, copied it in my own teaching and staff development work. But always on paper. For very small children, a paper sheet is the only approach that works (mark each correctly entered number with a brightly coloured counter or, if appropriate in the context, a sweet or piece of dried fruit). For older pupils, however, hands on ICT approaches offer tremendous potential - and Allmath.com’s interactive “sheet of paper equivalent” (see below) is wonderful. The missing element has, until now, been an instant, hands on generator and explorer of any n×n magic square or squares on demand.

For the teacher, Matlab and many compatible systems (including the free version of Sysquake and its Palm implementation Lyme) offer a very useful command to generate magic squares: “magic(n)” where n is the size of the square. (My thanks to AbsentCat, who pointed me to these resources.) For some older pupils, these are also useful.

There are a lot of useful materials on the web for building an ICT based “magic square portal” in the classroom. All that is needed is an interactive square calculator. For older secondary ages (Y8 for some pupils, Y13 or beyond for others), Sysquake Remote web implementation is a possibility, but not for the primary years. The Wolfram Demonstrations Project and free player, however, offer just the thing: a magic square generator with “dragable” column/row/locus cursor.

This Mathematica demonstration allows a magic square of any (odd number) size from 1 to 13 to be generated instantly using a slider at the top of the frame. A cursor can then be dragged around the square, highlighting the row and column containing a particular selected cell. Computation is left to the pupil, which is valuable arithmetic practice, but the cells involved are clearly isolated which minimises mistakes. A perfect fit for the missing piece in the ICT magic squares session.

Starting points for other material which has served me well are:

• Allan Adler’s Mathforum pages on magic squares
• Allmath.com’s interactive equivalent of a paper magic square sheet

[contributed by Chandra]

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]

Learning to start small in Cabri3D

I was interested in AbsentCat’s Active Geometry post, talking of a “geometry processor” doing wonderful things, but learning to use it sounded too much to cope with in the endless pressure of a school day. I work with young teenagers who should be in early Key Stage Four, but, because their previous education has been disrupted by events beyond their control or a teacher’s capacity to imagine, are in most cases struggling to master KS2 or even KS1. How could this active geometry business help them or me? But, during a staff development workshop, I was shown the ready made examples accompanying a copy of the three dimensional version, Cabri3D, mentioned by Philip Yorke. One of them addressed a topic which I was due to tackle with my youngsters: the nets of a solid. When I expressed interest in that example, I was offered a short loan of the laptop on which the workshop demonstration had been run.

Very uncertainly, I rigged up the machine and waited for my class. And they loved it. They are very videogame savvy, and related to the direct manipulation of an onscreen object in a way they had never related to paper or cardboard equivalents. The software allows them to pick up a single section of the flattened net, swing it along a guide path, and have the attached panels follow it. The cube assembles itself. Then a face can be swung back, restoring the 3D solid to flat net. As many times as they wish. With the ice broken, they were then amenable to physical exploration in a way that they had never been before.

Since then, I have read Chandra’s account of her Beanbag Thrower - another example of starting small, with a simple and manageable aspect, not being overawed by the greater power available. I have learned a lot from the experience, and learned a lot too from her account. Next time, I shall be bolder - integrating the software into a lesson plan, as she has, rather than just starting with it. I have borrowed the demonstration machine again, and next week I shall be using the 2D version, Cabri II Plus, for a more ambitious project around bicycle wheels. I intend to watch the demonstration videos, have signed up for another workshop on geometric software in CDT, and plan to spend the summer reinventing myself - bring it on.

[contributed by BobTheBumbler]