Just like that…

Responsiveness in a software publisher or supplier is always welcome, and more common than is often realised, but I’ve just experienced a particularly impressive case.On Thursday, I downloaded a 30 day trial copy of FX Draw 3 - part of FX MathPack, a suite of mathematics resources for the secondary education market by Australian publisherEfofex and available in the UK from Chartwell-Yorke).I had in mind buying a copy for use on a laptop with all the neighbourhood teenagers who drop in hopefully for assistance with homework, and in the café based outreach work with which I’m involved. Since the next topic likely to come up is bearings, I skipped to the angle measure resources and specifically to the onscreen protractors.There are two protractors: 180° and 360°. Both are very flexible and intuitive to use: they can be drawn on the fly, during an explanation, in less than a second, with one flick of the mouse. For bearings work, the 360° is by far the preferable one … but bearings are measured clockwise from north, and this protractor showed the full three hundred and sixty degrees only in the counterclockwise direction. After fiddling and exploring on my own for a bit, I emailed a query to Chartwell-Yorke.On Friday I received a reply from Efofex: no, there was no clockwise scale, but they had looked at the issue and would add one.Today, Sunday, another Efofex email dropped into my inbox: a clockwise scale on 360 degree protractors has been added to FX Draw and will appear in the next release (no date as yet for that release, which has to include other developments, but it’s expected “in a few weeks”.)You really can’t fault that.[contributed by Felix Grant]

Netbooks on the road

My part of this “netbooks” trial involved much hair loss. Since the base for my work with disconnected teenagers is a cybercafé, there is no obvious rôle for a small, pocketable computer in the normal context of what I do. To make good use of the opportunity, I had to let these machines go out of my control, into an environment where small high value objects are regarded as currency. The sponsors said they were willing to take the risk of loss, provided that I took what I considered reasonable care to minimise it … what, exactly, constitutes reasonable care when handing expensive stuff over to teenagers who may not come back, have class A drug habits, and are due in court on Wednesday for handling stolen goods?

The other question was what exactly to do with these machines, to justify taking the risk. These two issues were linked; my clients had to feel that something worthwhile was going on, if they were to respect the tools involved.

One subject which interests all of them, regardless of gender, is cars. A month before the netbooks arrived, I started discussing with them the relationships between weight, power, speed and acceleration in a car. They have rather more practical understanding of these matters than can be easily explained by legal experience at their age so I concentrated on trying to relate this to theoretical engineering models, first visual and then symbolic.

With the netbooks on hand, I brought the talk around to how we might investigate the actual (rather than maximum or advertised) speed and acceleration values for real cars in daily use. They were very interested in this idea, and were keen to try their hand at using spreadsheets for the purpose. Then they realised that they would have to write down a lot of information and bring it back to the centre, then key it in, before they could do anything with it; at that point, disappointment and loss of interest threatened. Like a good conjuror, I then produced the netbooks.

Gathering data

The scheme they devised involved teams of six, each team stationed downstream from a Pedestrian Light Controlled crossing (this allowed two teams per crossing, getting double data for each red light, at three different crossings). The team leader (let’s call her or him “A”) would stand by the lights themselves, and would have the computer with an open spreadsheet. “B” through to “F” would be at measured distances downstream from the lights.

When the lights turned red (probably because “A” had pressed the button, but I didn’t enquire too closely), “A” would take up a position beside the frontmost car and enter details (make, model including engine size if possible, number of occupants) into the spreadsheet. When the lights went amber, “A” would raise his or her arm and the others would prepare to start stopwatches (mostly on mobile phones, though a few used the function on their wristwatches). When the lights turned green “A” would drop the raised arm and start walking up the line; the rest of the team would start the stopwatches running.

As the lead car passed each team member, the stopwatch at that position would be stopped. As “A” reached each, the time on their stop watch would be entered into the spreadsheet. In this way, a database of timings at fixed distances for different vehicles was built up. The results were also visible in a predefined scatter plot at the right of the same screen, with an interpolated trend line, so the model could be seen developing as they worked. When complete, the sets of data were merged into a single sheet on the desk top and then filtered to compare different data for similar subsets.

As for the risk, I handed over the complete trial set to the two alpha primes in the group (one male, one female) and left them to arrange distribution; and all came back.

Taking it further

This probably seems an underutilisation of the equipment. The same data collection could, after all, have been done with a pocket PC or similar (in fact, the idea was partly suggested by Chandra’s Big Freeze which used Psion clamshells. But the experience of taking “proper computers” out, and being trusted to do so, was worth its weight in gold and stimulated desire to learn. There were, in any case, two follow ups which would not have been possible with handhelds.

First, there was use of a pure mathematics package to compare the experimental data with a theoretical model. Chandra and AbsentCat had described their use of SysQuake LE for projectile modelling. SysQuake is available for both Windows (in the cybercafé) and Linux (on the netbooks) so I installed both. Having set up a basic acceleration equation (d=½at²) on the PC, we set the value of a by trial and error to give a line which matched the spreadsheet data. The young people found this very empowering, and probably learnt more algebraic confidence in half an hour of SysQuake than in all of their time with me to date. They also learned, to their surprise, that most acceleration is over within a very short time (with speed surprisingly low and surprisingly constant) on urban roads.

Second, AbsentCat scrounged us the loan of a set of plug in USB interfaces allowing various types of switch to start or stop timers on the netbooks. The students had a lot of fun with trying out various switching devices. We were loaned some pressure mats which could be placed on the road, though too often the passing vehicles avoided them. We experimented with home made trembler switches, but they were too sensitive, and hard to position usefully. Lengths of rubber tube, filled with water, were laid across the road with light pressure sensitive microswitches plugged into the ends – these were the most successful, and supplied 95% of our usable data.

Broader benefits

The tremblers were a complete failure in data collection terms but worth their weight in gold for the interest which they provoked. A drop of mercury is placed in the bottom of a glass tube; one electrode is immersed in it, and another arranged as a circular collar around the inside of the tube, fractionally above the meniscus; any motion which shakes the tube causes the mercury to make contact between the two electrodes, completing a circuit. Most of my clients have, at some time, been involved in vehicle theft, and immediately realised the relevance of tremblers to car alarms. We got a lot of chemistry, physics and engineering time out of the resulting investigations – even starting a new set of data collection exercises to investigate the link between tube size, collar spacing, and the trade off between sensitivity and discrimination.

This second (more accurate) phase gave us enough data to further investigate the mathematical model, and to extend it into areas such as mechanical work or power/weight ratios. It also allowed us to compare vehicles by type (small car, four wheel drive, bus, lorry, motorcycle, etc). Most valuably, in some ways, it led on naturally to discussing the range of road behaviours exhibited by different users of the same vehicle.

[Contributed by BobTheBumbler]

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]

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]

Graph magic!

Magic is a stage in the developmental history of science — a history which each of us retraces as we grow to intellectual maturity. Its study as such by eight-year olds was designed to meet criteria in cultural history and imaginative creation, but also as a context for strengthening critical faculties. The ability to rationally assess likely and implausible explanations of phenomena makes great strides at this age; separation of reality from model is central.

A link between mathematical models and spells, illustrated by Omnigraph, was well received and opened up a riot of speculative theorising. It also offered a new stage on which to parade the key concept of the algebraic “placeholder”.

Omnigraph is a graph processor, with facilities for investigating a number of mathematical areas up to very basic calculus. Equations or Cartesian coordinates, entered from keyboard or menus, are instantly reflected in curves, lines, points and shapes drawn in the graph window. Or, looked at another way, “spells” in the lower window produce magical results in the upper one - but rules can be deduced, even at this age, to predict the result of any given spell.

The mouse changes scaling, draws tangents, normals, areas, and the rest; curve drawing can be paused or abandoned, and in many cases the equation/spell is displayed as the mouse passes over a line.

A quadratic spell produces a passable model of the path followed by Harry Potter’s broomstick as he swoops to aid a Quidditch team-mate before returning to his normal altitude position. We can also play the part of the villainous Quirrel, interfering with the spell to alter Harry’s flight: alter one part of the spell (the m coefficient) to induce suicidal recklessness; change another (the constant c) to pull him out of the dive earlier - or cause him to crash!

If it looks like I’m getting carried away - well, perhaps I am. There is nothing more inspiring than watching young minds leap over their fears and years to grasp an idea. By the end of the morning, any member of the class could evaluate the value of y for any x, plotting the results on a graph paper Quidditch field. They could also deal implicitly with negative values for m and c, expressed as subtractions in a modified “spell”.

We assembled a tolerable Cartesian cartoon representation of Nearly Headless Nick, behind a transparent acetate screen overlay carrying a Hogwarts map. The pupils derived great amusement and insight from altering transformation matrix-spells to move Nick about the castle, expand him, shrink him, distort him in various ways…

Omnigraph is a simple, no frills program in its interaction with the user, which makes it very transparent in use. It is also well known; all the teachers involved had encountered it, if not used it, before. For more advanced work it could be replaced by Autograph; this would sacrifice instant usability in favour of added options. Both programs work well in conjunction with graphical calculators, for teaching at the levels where those are appropriate. Autograph offers stronger tools (eigenvalues, for instance), enhanced display options and statistical data plotting.

[Contributed by AbsentCat]