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]

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]

Virtual experiments from Kinetic Books

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[Contributed by AbsentCat]