Pick a Scale

Back in time, when I was still a salaried academic, one of the students in the course on Matter I taught [1] managed to derive a novel form of creativity from my lectures. I habitually made audio recordings – later, video recordings – of my lectures available to students for revision purposes (and to help with dyslexia etc; see here and other related posts) but this one student found another use for them. He cut snippets from my lecture recordings and dubbed them onto a piece of music. I only found out about it after it had travelled, viral fashion, around the student body. I took it as a compliment, and still do. In order that you can enjoy this as well I've uploaded it to YouTube, here: it benefits from volume and decent bass [2]. One of several things I learnt from his creativity is that I had, and most probably still have catch-phrases. There was one, however, that almost certainly led the pack in the context of this lecture course: “pick a scale” and its close variants.

Why? Because the most common cause of needless mistakes within their numerical exercises was the admixture of measurement scales and a confusion regarding the units associated with a given scale. Thus, someone might mix grams with kilograms and be out by a factor of 1000 – make this mistake more than once in a calculation, or do so in the context of an equation in which the mass, for example, appears more than once, and the results will be thrown even wider off the mark. It was a problem that dogged the students in this programme more than most simply because they were often older than the usual direct-entry undergraduates and/or came from a wider range of educational backgrounds. Many of them had, like me, grown up with measurement scales and units that were commonly used before SI units, the système international d'unités, held sway as our metric framework.

A ‘useful’ plastic ruler from my past: upper scale showing centimetres and their metric sub-division into millimetres; lower scales showing inches and subdivisions into tenths, twelfths and sixteenths.

Within my own formal education I had begun with imperial units (miles, feet & inches; pounds, hundredweight and tons; hours; degrees Fahrenheit …) and all the derived units that went with them or alongside them – like foot-pounds to quantify energy and pounds per square inch as the unit of pressure. Many of us in the UK will still think in those terms on a day-to-day basis; many more in the USA will follow suit. I was versed in these things until my mid-teens; even the wicket on the cricket pitches my father and brother played on, and my son still does, are precisely one chain long (stumps to stumps; one chain = 22 yards = 66 feet). However, by then I was veering towards the sciences, and therefore also mathematics, and metrication became the thing – we were required to familiarise ourselves with the CGS system: centimetres, grams and seconds, and associated units like degrees centigrade. We were obliged to change yet again within two or three years. By then I was in the final stages of my secondary schooling – senior years of High School within the USA, approximately – and had opted to specialise solely in the physical sciences. (I specialised because that was what one did back then, and it remains the norm in England today unfortunately. By preference I would have added English Literature and either Archaeology, Logic/Philosophy to the mix.) This time it was a blessedly less radical move into the MKS system: metres, kilograms and seconds, and their associated units. The final change, or at least I hope and believe it’s the final one, came when I became an undergraduate Physics student: MKS moved almost effortlessly into the SI system, which retains the same base units of length, mass and time.

My wife still uses these imperial-scale scales in the kitchen, as do I when following classic old recipes – particularly for jam (or jelly for those in the USA). The mass currently on the scales is one pound (1 lb); the others shown are ½ lb, 4 ounces (4 oz, i.e. ¼ lb), 2 oz and 1oz.

Thus, I could empathise with my students because I had seen even larger shifts in the way we quantify our measurements of our world than they had. However, empathy alone was of little benefit to them – hence the need for many reminders to make sure all the numbers they were using were associated with the same measurement system, preferably SI: whence was born a catch-phrase …

Lest anyone think this is an issue of minor irritation to a few early-years students, I’ll share with you the story of NASA’s $125M Mars Climate Orbiter from the late ‘90s. It burned up in the Martian atmosphere because engineers had failed to convert units from imperial to metric (see here). To bring it forward to today, just imagine the consequences of an analogous error in the software of a driverless vehicle. Scales and units are a non-trivial issue.

Mars Climate Orbiter 

Nothing in what I have said should be taken as a statement that the scales and units of former days were intrinsically inferior. I habitually use the metric system – almost always the SI – because it makes my life easier, both as a scientist and when in the kitchen or at my workbench, but these other scales have their strengths. Imperial scales are often quite intuitive for instance: a foot corresponds to just that – the average length of an adult foot, and an adult’s stride becomes a yard; the acre, a measure of area, was defined in terms of the farmland that might reasonably be ploughed in one day prior to mechanisation. The Fahrenheit scale, likewise, was established such that 0ºF corresponds to the lowest temperatures one might expect in Winter (- bearing in mind the parts of the globe in which the scale was being used) and 100ºF to the highest Summer temperatures to be expected. It’s all very sensible as far as it goes. Moreover, even now, and amongst a scientific community using SI units by default, there are ‘useful exceptions’. Astronomers and planetary scientists speak in terms of the astronomical unit, AU, for example, which is simply the average Sun-Earth distance; mass is often given in units of Earth’s mass or the Sun’s mass. Closer to home, although I perhaps ought to be employing the nanometre, nm, I still use a length called the Ångström when I’m discussing the separations between atoms: this was initially established as the diameter of a hydrogen atom – the smallest atom and therefore a useful ‘measuring stick’ in this realm. There are 10 Å in each nm.

For a useful compilation of scales and units, old and new, see here.

An obvious weakness in the historical scales, certainly as they were originally established, is that they could – and did – vary quite markedly from one community to another, much as the time of day varied within a country when all calibration derived from the Sun. Even after standardization was achieved there remained significant issues, not least with the English-speaking world. For instance, a gallon in the UK is not the same volume as a gallon in the USA: in fact it’s approximately 20% larger. However, these are well understood differences. More troublesome by far, to my mind at least, are those ad hoc scales/units invented ostensibly to help us understand something but which serve only to obfuscate and confuse. There is a depressingly long list of examples: measuring an area using a ‘football pitch’ as the unit, or a height in terms of Nelson’s Column in London, or a length in units of London buses, … My personal ‘favourite’, which I spotted in one of my grandsons’ books a few years ago, came from an author trying to convey the mass of one of the monoliths which make up Stonehenge by informing us that is was the same as 22 sheep! I can’t help but think that these inventions are counter-productive.


Footnotes
[1] I was teaching this within our Foundation Year programme which I helped to start back in the early ‘90s. The programme, inserted before the more conventional undergraduate degree programme, has offered a ‘second chance’ to hundreds of students in the succeeding years. I loved teaching within it, and the students I met whilst doing it.
It’s amusing to see that the lead text at the top of the current web page still has its roots in what I wrote for printed course booklets back then: this Physics evidently ages well.

[2] The music is Tractor Beam by Eat Static, which is used with their kind permission; the image is of Bulkhead by Rick Kirby and this stands outside the Marlowe Theatre, Canterbury, UK.

Image of Stonehenge adapted from http://www.english-heritage.org.uk/visit/places/stonehenge/ and the cartoon sheep are extracted from http://clipart-library.com/clipart/1686234.htm