## A Gnome-inal Value for ‘g’

. . . setting storms and billows at defiance, and visiting the remotest parts of the terraqueous globe.

Samuel Johnson, The Rambler, 17 April 1750

That an object in free fall will accelerate towards the centre of our terraqueous globe at a rate of 9.81 metres per second per second is, at best, only a partial and parochial truth. It is 9.81 metres per second per second in the United Kingdom, yes; but the value of both acceleration due to free fall and the gravitational field strength vary from place to place across the globe (and in the SI System of measurement, the two quantities are numerically equal and dimensionally equivalent).

For example, according to Hirt et al. (2013) the lowest value for g on the Earth’s surface is atop Mount Huascarán in Peru where g = 9.7639 m s-2 and the highest is at the surface of the Arctic Ocean where g = 9.8337 m s-2.

### Why does g vary?

There are three factors which can affect the local value of g.

Firstly, the distribution of mass within the volume of the Earth. The Earth is not of uniform density and volumes of rock within the crust of especially high or low density could affect g at the surface. The density of the rocks comprising the Earth’s crust varies between 2.6 – 2.9 g/cm3 (according to Jones 2007). This is a variation of 10% but the crust only comprises about 1.6% of the Earth’s mass since the density of material in the mantle and core is far higher so the variation in g due this factor is probably of the order of 0.2%.

Secondly, the Earth is not a perfect sphere but rather an oblate spheroid that bulges at the equator so that the equatorial radius is 6378 km but the polar radius is 6357 km. This is a variation of 0.33% but since the gravitational force is proportional to 1/r2 let’s assume that this accounts for a possible variation of the order of 0.7% in the value of g.

Thirdly, the acceleration due to the rotation of the Earth. We will look in detail at the theory underlying this in a moment, but from our rough and ready calculations above, it would seem that this is the major factor accounting for any variation in g: that is to say, g is a minimum at the equator and a maximum at the poles because of the Earth’s rotation.

### The Gnome Experiment

In 2012, precision scale manufacturers Kern and Sohn used this well-known variation in the value of g to embark on a highly successful advertising campaign they called the ‘Gnome Experiment’ (see link 1 and link 2).

Whatever units their lying LCD displays show, electronic scales don’t measure mass or even weight: they actually measure the reaction force the scales exert on the item in their top pan. The reading will be affected if the scales are accelerating.

In diagram B, the apple and scales are in an elevator that is accelerating upward at 1.00 metres per second per second. The resultant upward force must therefore be larger than the downward weight as shown in the free body diagram. The scales show a reading of 1.081/9.81 – 0.110 194 kg = 110.194 g.

In diagram C, the the apple and scales are in an elevator that is accelerating downwards at 1.00 metres per second per second. The resultant upward force must therefore be smaller than the downward weight as shown in the free body diagram. The scales show a reading of 0.881/9.81 – 0.089 806 kg = 89.806 g.

### Never mind the weight, feel the acceleration

Now let’s look at the situation the Kern gnome mentioned above. The gnome was measured to have a ‘mass’ (or ‘reaction force’ calibrated in grams, really) of 309.82 g at the South Pole.

Showing this situation on a diagram:

Looking at the free body diagram for Kern the Gnome at the equator, we see that his reaction force must be less than his weight in order to produce the required centripetal acceleration towards the centre of the Earth. Assuming the scales are calibrated for the UK this would predict a reading on the scales of 3.029/9.81= 0.30875 kg = 308.75 g.

The actual value recorded at the equator during the Gnome Experiment was 307.86 g, a discrepancy of 0.3% which would suggest a contribution from one or both of the first two factors affecting g as discussed at the beginning of this post.

Although the work of Hirt et al. (2013) may seem the definitive scientific word on the gravitational environment close to the Earth’s surface, there is great value in taking measurements that are perhaps more directly understandable to check our comprehension: and that I think explains the emotional resonance that many felt in response to the Kern Gnome Experiment. There is a role for the ‘artificer’ as well as the ‘philosopher’ in the scientific enterprise on which humanity has embarked, but perhaps Samuel Johnson put it more eloquently:

The philosopher may very justly be delighted with the extent of his views, the artificer with the readiness of his hands; but let the one remember, that, without mechanical performances, refined speculation is an empty dream, and the other, that, without theoretical reasoning, dexterity is little more than a brute instinct.

Samuel Johnson, The Rambler, 17 April 1750

### References

Hirt, C., Claessens, S., Fecher, T., Kuhn, M., Pail, R., & Rexer, M. (2013). New ultrahigh‐resolution picture of Earth’s gravity fieldGeophysical research letters40(16), 4279-4283.

Jones, F. (2007). Geophysics Foundations: Physical Properties: Density. University of British Columbia website, accessed on 2/5/21.

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## Mnemonics for the S.I. Prefixes

The S.I. System of Weights and Measures may be a bit of a dog’s dinner, but at least it’s a dog’s dinner prepped, cooked, served and — more to the point — eaten by scientists.

### A brief history of the Système international d’unités

It all began with the métre (“measure”), of course. This was first proposed as a universal measure of distance by the post-Revolutionary French Academy of Sciences in 1791. According to legend (well, not legend precisely — think of it as random speculative gossip, if you prefer), they first proposed that the metre should be one millionth of the distance from the North Pole to the equator.

When that turned out to be a little on the large side, they reputedly shrugged in that inimitable Gallic fashion said: “D’accord, faisons un dix millionième alors, mais c’est ma dernière offre.” (“OK, let’s make it one ten millionths then, but that’s my final offer.”)

Since then, what measurement-barbarians loosely (and egregiously incorrectly) call “the metric system” has been through many iterations and revisions to become the S.I. System. Its full name is the Système international d’unités which pays due honour to France’s pivotal role in developing and sustaining it.

When some of those same measurement-barbarians call for a return to the good old “pragmatic” Britsh Imperial System of inches and foot-poundals (the force exerted by a mass of one pound dropped from a height of one foot, in case you didn’t know), I urge all fair-minded people to tell them, as kindly as possible, that they can’t: not now, not ever.

Since 1930, the inch has been defined as 25.4 millimetres. (It was, so I believe, the accuracy and precision needed to design and build jet engines that led to the redefinition. The older definitions of the inch simply weren’t precise enough.)

You simply cannot replace the S.I. system, you can, however, dress it up a little bit and call a distance of 25.4 millimetres “one inch” if you really wanted to — but, in the end, what would be the point of that?

### The Power of Three (well, ten to the third power, anyways)

For human convenience, the S.I. system includes prefixes. So a large distance might measured in kilometres where the prefix kilo- indicates multiplying by a factor of 1000 (or 10 raised to the third power). The distance between the Globe Theatre in London and Slough Station is 38.6 km. Longer distances such as London and New York, NY would be 5.6 megametres (or 5.6 Mm — note capital ‘M’ for mega [one million] to avoid confusion with the prefix milli- ).

The S.I. System has prefixes for all occasions, as shown below.

Note also that one should convert all prefixes into standard units for calculations e.g. meganewtons should be converted to newtons. The sole exception is kilograms because the base unit is the kilogram not the gram, so a megagram should be converted into kilograms, not grams. I trust that’s clear. (Did I mention the “dog’s dinner” part yet?)

For perspective, the distance between Earth and the nearest star outside our Solar System is 40 petametres, and current age of the universe is estimated to be 0.4 exaseconds (give or take a petasecond or two).

A useful mnemonic for remembering these is Karl Marx Gives The Proletariat Eleven Zeppelins (and one can imagine the proletariat expressing their gratitude by chanting in chorus: “Yo! Ta, Mr Marx!” as they march bravely forward.)

But what about the little prefixes?

Milli- we have already covered above. The diameter of one of your red blood cells in 8 micrometres and the time it takes light to travel a distance equal to the diameter of a hydrogen atom is 300 zeptoseconds.

Again, there is an SI prefix for every occasion:

A useful mnemonic would be: Millie’s Microphone Needs a Platform For Auditioning Zebra Yodellers.

For the record, GCSE Physics students are expected to know the SI prefixes between giga- and pico-, but if you’re in for a pico- then you’re in for anything between a yotta- and a yocto- in my opinion (if you catch my drift).

### Very, very, very small to very, very, very big

The mean lifetime of a Z-boson (the particle that carries the Weak force) is 0.26 yoctoseconds.

According to our current understanding of physics, the stars will have stopped shining and all the galaxies will dissipate into dissassociated ions some 315 yottaseconds from now.

Apart from that, happy holidays everyone!

## Reducing Cognitive Overload in Practicals by graphing with Excel

Confession, they say, is good for the soul. I regret to say that for far too many years as a Science teacher, I was in the habit of simply ‘throwing a practical’ at a class in the belief that it was the best way for students to learn.

However, I now believe that this is not the case. It is another example of the ‘curse of the expert’. As a group, Science teachers are (whether you believe this of yourself and your colleagues or not) a pretty accomplished group of professionals. That is to say, we don’t struggle to use measuring instruments such as measuring cylinders, metre rules (not ‘metre sticks’, please, for the love of all that’s holy), ammeters or voltmeters. Through repeated practice, we have pretty much mastered tasks such as tabulating data, calculating the mean, scaling axes and plotting graphs to the point of automaticity.

But our students have not. The cognitive load of each of the myriad tasks associated with the successful completion of full practical should not be underestimated. For some students, it must seem like we’re asking them to climb Mount Everest while wearing plimsols and completing a cryptic crossword with one arm tied behind their back.

One strategy for managing this cognitive load is Adam Boxer’s excellent Slow Practical method. Another strategy, which can be used in tandem with the Slow Practical method or on its own, is to ‘atomise’ the practical and focus on specific tasks, as Fabio Di Salvo suggests here.

### Simplifying Graphs (KS3 and KS4)

If we want to focus on our students’ graph scaling and plotting skills, it is often better to supply the data they are required to plot. If the focus is interpreting the data, then Excel provides an excellent tool for either: a) providing ready scaled axes; or b) completing the plotting process.

Typical exam board guidance states that computer drawn graphs are acceptable provided they are approximately A4 sized and include a ‘fine grid’ similar to that of standard graph paper (say 2 mm by 2 mm) is used.

Excel has the functionality to produce ‘fine grids’ but this can be a little tricky to access, so I have prepared a generic version here: Simple Graphs workbook link.

Data is entered on the DATA1 tab. (BTW if you wish to access the locked non-green cells, go to Review > Unlock sheet)

The data is automatically plotted on the ‘CHART1 (with plots)’ tab.

Please note that I hardly ever use the automatic trendline drawing functionality of Excel as I think students always need practice at drawing a line of best fit from plotted points.

Alternatively, the teacher can hand out a ‘blank’ graph with scaled axes using the ‘CHART1 (without) plots’ tab.

### Using the Simple Graph workbook with a class

I have used this successfully with classes in a number of ways:

• Plotting the data of a demo ‘live’ and printing out a copy of the completed graph for each student.
• Supplying laptops or tablet so that students can enter their own data ‘live’.
• Posting the workbook on a VLE so that students can process their own data later or for homework.

### Adjusting the Simple Results Graph workbook for different ranges

But what if the data range you wish to enter is vastly different from the generic values I have randomly chosen?

It may look like a disaster, but it can be resolved fairly easily.

Firstly, right click (or ctrl+click on a Mac) on any number on the x-axis. Select ‘Format Axis’ and navigate to the sub-menu that has the ‘Maximum’ and ‘Minimum’ values displayed.

Since my max x data value is 60 I have chosen 70. (BTW clicking on the curved arrow may activate the auto-ranging function.)

I also choose a suitable value of ’10’ for the “Major unit’ which is were the tick marks appear. And I also choose a value of ‘1’ for the minor unit (Generally ‘Major unit’/10 is a good choice)

Next, we right click on any number on the y-axis and select ‘Format Axis’. Going through a similar process for the y-axis yields this:

… which, hopefully, means ‘JOB DONE’

### Plotting More Advanced Graphs at KS4 and KS5

The ‘Results Graph (KS4 and KS5)’ workbook (click on link to access and download) will not only calculate the mean of a set of repeats, but will also calculate absolute uncertainties, percentage uncertainties and plot error bars.

Again, I encourage students to manually draw a line of best fit for the data, and (possibly) calculate a gradient and so on.

### And finally…

Happy graphing, folks 🙂

## Magnetism? THERE IS NO MAGNETISM!!!!

Has a school physics experiment or demonstration ever changed the course of human history?

On 21 April 1820, one such demonstration most definitely did. According to physics lore, Hans Christian Øersted was attempting to demonstrate to his students that, according to the scientific understanding of the day, there was in fact no connection between magnetism and electricity.

To this laudable end, he placed a compass needle near to a wire to show that when the current was switched on, the needle would not be affected.

Except that it was affected. Frequently. Each and every time Øersted switched on the electric current, the needle was deflected from pointing North.

Everybody has heard that wise old saw that ‘If it doesn’t work, it’s physics…” except that in this case ‘It did actually work as it was supposed to but in an unexpected way due to a hitherto-unknown-completely-new-branch-of-physics.’

Øersted, to his eternal credit, did not let it lie there and was a pioneer of the new science of electromagnetism.

### Push-me-pull-you: or, two current-carrying conductors

One curious consequence of Øersted’s new science was the realisation that, since electric currents create magnetic fields, two wires carrying electric currents will exert a force on each other.

Let’s consider two long, straight conductors placed parallel to each other as shown.

In the diagram above, the magnetic field produced by the current in A is shown by the green lines. Applying Fleming’s Left Hand Rule* to conductor B, we find that a force is produced on B which acts towards conductor A. We could go through a similar process to find the force acting on B, but it’s far easier to apply Newton’s Third Law instead: if body A exerts a force on body B, then body B exerts an equal and opposite force on body A. Hence, conductor A experiences a force which pulls it towards conductor B.

So, two long, straight conductors carrying currents in the same direction will be attracted to each other. By a similar analysis, we find that two long, straight conductors carrying currents in opposite directions will be repelled from each other.

In the past, this phenomenon was used to define the ampere as the unit of current: ‘The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 m apart in vacuum, would produce between these conductors a force equal to 2×10−7 newton per metre of length.‘ However, the 2019 redefinition of the SI system has ditched this and adopted a new definition in terms of the transfer of the elementary charge, e.

### Enter Albert Einstein, pursuing an enigma

What is the connection between magnetism and electricity? It was precisely this puzzle that started Albert Einstein on the road to special relativity. It is one of the unsung triumphs of this theory that it lays bare the connection between magnetism and electricity.

In what follows, we’re going to apply Einstein’s analysis to the situation of two long, straight current-carrying conductors. Acknowledgment: I’m going to following a line of argument laid out in Beiser 1988: 19-22.

### It’s gotta be perfect (or ‘idealised’, if you prefer)

Let’s consider two idealised conductors A and B both at rest in the inertial reference frame of the laboratory. The flow of charge in both conductors is made up of positive and negative charge carriers moving in opposite directions with a speed v.

None of the charges in A interact with the other charges in A because we are considering an idealised conductor. However, the charges in A will interact with the charges in B.

### Flip the inertial reference frame

Now let’s look at the situation from the inertial reference frame of one of the positive charges in A. For simplicity, we can focus on a single positive charge in A since it does not interact with any of the other charges in A.

With reference to this inertial frame, the positive charge in A is stationary and the positive charges in B are also stationary.

However, the inertial frame of the laboratory is moving right-to-left with a speed v and the negative charges are moving right-to-left with a speed of 2v.

Since the positive charges in B are stationary with respect to the positive charge in A, the distance between them is the same as it was in the laboratory inertial frame. However, since the negative charges in B are moving with speed 2v with respect to positive charge in A, the spacing between is contracted due to relativistic length contraction (see Lottie and Lorentzian Length Contraction).

Because of this, the negative charge density of B increases since they are closer together. However, the positive charge density of B remains the same since they are stationary relative to the positive charge in A so there is no length contraction.

This means that, as far as the positive charge in A is concerned, conductor B has a net negative charge which means the positive charge experiences an attractive Coulomb’s Law electrical force towards B.

A similar analysis applied to electric currents in opposite directions would show that the positive charge in A would experience a repulsive Coulomb’s Law electrical force. The spacing between the positive charges in B would be contracted but the spacing between the negative charges remains unchanged, so conductor B has a net positive charge because the positive charge density has increased but the negative charge density is unchanged.

### Magnetism? THERE IS NO MAGNETISM!!!!

So what we normally think of as a ‘magnetic’ force in the inertial frame of the laboratory can be explained as a consequence of special relativity altering the charge densities in conductors. Although we have just considered a special case, all magnetic phenomena can be interpreted on the basis of Coulomb’s Law, charge invariance** and special relativity.

For the interested reader, Duffin (1980: 388-390) offers a quantitative analysis where he uses a similar argument to derive the expression for the magnetic field due to a long straight conductor.

Update: I’m also indebted to @sbdugdale who points out the there’s a good treatment of this in the Feynman Lectures on Physics, section 13.6.

### Notes and references

* Although you could use a non-FLHR catapult field analysis, of course

** ‘A current-carrying conductor that is electrically neutral in one frame of reference might not be neutral in another frame. How can this observation be reconciled with charge invariance? The answer is that we must consider the entire circuit of which the conductor is a part. Because a circuit must be closed for a current to occur in it, for every current element in one direction that a moving observer find to have, say, a positive charge, there must be another current element in the opposite direction which the same observer finds to have a negative charge. Hence, magnetic forces always act between different parts of the same circuit, even though the circuit as a whole appears electrically neutral to all observers.’ Beiser 1988: 21

Beiser, A. (1988). Concepts of modern physics. Tata McGraw-Hill Education

Duffin, W. J. (1980). Electricity and magnetism. McGraw-Hill.

## The Acceleration Required Practical Without Light Gates (And Without Tears)

### Introduction

The AQA GCSE Required Practical on Acceleration (see pp. 21-22 and pp. 55-57) has proved to be problematic for many teachers, especially those who do not have access to a working set of light gates and data logging equipment.

In version 3.8 of the Practical Handbook (pre-March 2018), AQA advised using the following equipment featuring a linear air track (LAT). The “vacuum cleaner set to blow”, (or more likely, a specialised LAT blower), creates a cushion of air that minimises friction between the glider and track.

However, in version 5.0 (dated March 2018) of the handbook, AQA put forward a very different method where schools were advised to video the motion of the car using a smartphone in an effort to obtain precise timings at the 20 cm, 40 cm and other marks.

It is possible that AQA published the revised version in response to a number of schools contacting them to say. “We don’t have a linear air track. Or light gates. Or a ‘vacuum cleaner set to blow’.”

The weakness of the “new” version (at least in my opinion) is that it is not quantitative: the method suggested merely records the times at which the toy car passed the lines. Many students may well be able to indirectly deduce the relationship between resultant force and acceleration from this raw timing data; but, to my mind, it would be cognitively less demanding if they were able to compare measurements of resultant force and acceleration instead.

### Adapting the AQA method to make it quantitative

We simplify the AQA method as above: we simply time how long the toy car takes to complete the whole journey from start to finish.

If a runway of one metre or longer is set up, then the total time for the journey of the toy car will be 20 seconds or so for the smallest accelerating weight: this makes manual timing perfectly feasible.

Important note: the length of the runway will be limited by the height of the bench. As soon as the weight stack hits the floor, the toy car will no longer experience an accelerating force and, while it may continue at a constant speed (more or less!) it will no longer be accelerating. In practice, the best way to sort this out is to pull the toy car back so that the weight stack is just below the pulley and mark this position as the start line; then slowly move the toy car forward until the weight stack is just touching the floor, and mark this position as the finish line. Measure the distance between the two lines and this is the length of your runway.

In addition, the weight stack should feature very small masses; that is to say, if you use 100 g masses then the toy car will accelerate very quickly and manual timing will prove to be impossible. In practice, we found that adding small metal washers to an improvised hook made from a paper clip worked well. We found the average mass of the washers by placing ten of them on a scale.

### The Eleventh Commandment: Thou Shalt Not Confound Thy Variables!

To confirm the straight line and directly proportional relationship between accelerating force and acceleration, bear in mind that the total mass of the accelerating system must remain constant in order for it to be a “fair test”.

The parts of our system that are accelerating are the toy car, the string and the weight stack. The total mass of the accelerating system shown below is 461 g (assuming the mass of the hook and the string are negligible).

The accelerating (or resultant) force is the weight of 0.2 g mass on the hook, which0 can be calculated using W = mg and will be equal to 0.00196 N or 1.96 mN.

In the second diagram, we have increased the mass on the weight stack to 0.4 g (and the accelerating force to 0.00392 N or 3.92 mN) but note that the total mass of the accelerating system is still the same at 461 g.

In practice, we found that using blu-tac to stick a matchbox tray to the roof of the car made managing and transferring the weight stack easier.

Personal note: as a beginning teacher, I demonstrated the linear air track version of this experiment to an A-level Physics class and ended up disconfirming Newton’s Second Law instead of confirming it; I was both embarrassed and immensely puzzled until an older, wiser colleague pointed out that the variables had been well and truly confounded by not keeping the total mass of the accelerating system constant.

### What lies beneath: the Physics underlying this method

This can be considered as “deep background” rather than necessary information, but I, for one, consider it really interesting.

Acceleration is the rate of change of a rate of change. Velocity is the rate of change of displacement with time and acceleration is the rate of change of velocity.

Interested individuals may care to delve into higher derivatives like jerk, snap, crackle and pop (I kid you not — these are the technical terms). Jerk is the rate of change of acceleration and hence can be defined as (takes a deep breath) the rate of change of a rate of change of rate of change. More can be found in the fascinating article by Eager, Pendrill and Reistad (2016) linked to above.

But on a much more prosaic level, acceleration can be defined as a = (v – u) / t where v is the final instantaneous velocity, u is the inital instantaneous velocity and t is the time taken for the change.

The instantaneous velocity is the velocity at a momentary instant of time. It is, if you like, the velocity indicated by the needle on a speedometer at a single instant of time and is different from the average velocity which is calculated from the total distance travelled divided the time taken.

This can be shown in diagram form like this:

However, our experiment is simplified because we made sure that the toy car was stationary when the timer was zero; in other words, we ensured u = 0 m/s.

This simplifies a = (v – u) / t to a = v / t.

But how can we find v, the instantaneous velocity at the end of the journey when we have no direct means of measuring it, such as a speedometer or a light gate?

### No more jerks left to give

Let’s assume that, for the toy car, the jerk is zero (again, let me emphasize that jerk is a technical term defined as the rate of change of acceleration).

This means that the acceleration is constant.

This fact allows us to calculate the average velocity using a very simple formula: average velocity = (u + v) / t .

But remember that u = 0 so average velocity = v / 2 .

More pertinently for us, provided that u = 0 and jerk = 0, it allows us to calculate a value for v using v = 2 x (average velocity) .

The spreadsheet linked to above uses this formula to calculate v and then uses a = v / t.

### Using this in the school laboratory

This could be done as a demonstration or, since only basic equipment is needed, a class experiment. Students may need access to computers running the spreadsheet during the experiment or soon afterwards. We found that one laptop shared between two groups was sufficient.

First experiment (relationship between force and acceleration): set up as shown in the diagram. Place washers totalling a mass of 0.8 g (or similar) and washers totalling a mass of 0.2 g on the hook or weight stack. Hold the toy car stationary at the start line. Release and start the timer. Stop the timer. Input data into the spreadsheet and repeat with different mass on the hook.

It can be useful to get students to manually “check” the value of a calculated by the spreadsheet to provide low stakes practice of using the acceleration formula.

Second experiment (relationship between mass and acceleration). Keep the accelerating force constant with (say) 0.6 g on the hook or weight stack. Hold the toy car stationary at the start line. Release and start the timer. Stop the timer. Input data into the second tab on the spreadsheet and repeat with 100 g added to the toy car (possibly blu-tac’ed into place).

### Conclusion

This blog post grew in the telling. Please let me know if you try the methods outlined here and how successful you found them

### References

Eager, D., Pendrill, A. M., & Reistad, N. (2016). Beyond velocity and acceleration: jerk, snap and higher derivatives. European Journal of Physics, 37(6), 065008.

### Postscript

You can read Part Deux of this blogpost, which details an adaptation of this experiment to work with dynamics trolleys and other standard laboratory equipment.

## The Life and Death of Stars

Stars, so far as we understand them today, are not “alive”.

Now and again we saw a binary and a third star approach one another so closely that one or other of the group reached out a filament of its substance toward its partner. Straining our supernatural vision, we saw these filaments break and condense into planets. And we were awed by the infinitesimal size and the rarity of these seeds of life among the lifeless host of the stars. But the stars themselves gave an irresistible impression of vitality. Strange that the movements of these merely physical things, these mere fire-balls, whirling and traveling according to the geometrical laws of their minutest particles, should seem so vital, so questing.

Olaf Stapledon, Star Maker (1937)

And yet, it still makes sense to speak of a star being “born”, “living” and even “dying”.

We have moved on from Stapledon’s poetic description of the formation of planets from a filament of star-stuff gravitationally teased-out by a near-miss between passing celestial orbs. This was known as the “Tidal Hypothesis” and was first put forward by Sir James Jeans in 1917. It implied that planets circling stars would be an incredibly rare occurrence.

Today, it would seem that the reverse is true: modern astronomy tells us that planets almost inevitably form as a nebula collapses to form a star. It appears that stars with planetary systems are the norm, rather than the exception.

Be that as it may, the purpose of this post is to share a way of teaching the “life cycle” of a star that I have found useful, and that many students seem to appreciate. It uses the old trick of using analogy to “couch abstract concepts in concrete terms” (Steven Pinker’s phrase).

I find it humbling to consider that currently there are no black dwarf stars anywhere in the observable universe, simply because the universe isn’t old enough. The universe is merely 13.7 billion years old. Not until the universe is some 70 000 times its current age (about 1015 years old) will enough time have elapsed for even our oldest white dwarfs to have cooled to become a black dwarf. If we take the entire current age of the universe to be one second past midnight on a single 24-hour day, then the first black dwarfs will come into existence at 8 pm in the evening…

And finally, although to the best of our knowledge, stars are in no meaningful sense “alive”, I cannot help but close with a few words from Stapledon’s riotous and romantic imaginative tour de force that is yet threaded through with the disciplined sinews of Stapledon’s understanding of the science of his day:

Stars are best regarded as living organisms, but organisms which are physiologically and psychologically of a very peculiar kind. The outer and middle layers of a mature star apparently consist of “tissues” woven of currents of incandescent gases. These gaseous tissues live and maintain the stellar consciousness by intercepting part of the immense flood of energy that wells from the congested and furiously active interior of the star. The innermost of the vital layers must be a kind of digestive apparatus which transmutes the crude radiation into forms required for the maintenance of the star’s life. Outside this digestive area lies some sort of coordinating layer, which may be thought of as the star’s brain. The outermost layers, including the corona, respond to the excessively faint stimuli of the star’s cosmical environment, to light from neighbouring stars, to cosmic rays, to the impact of meteors, to tidal stresses caused by the gravitational influence of planets or of other stars. These influences could not, of course, produce any clear impression but for a strange tissue of gaseous sense organs, which discriminate between them in respect of quality and direction, and transmit information to the correlating “brain” layer.

Olaf Stapledon, Star Maker (1937)

## Teaching Magnification Using the Singapore Bar Model

He was particularly indignant against the almost universal use of the word idea in the sense of notion or opinion, when it is clear that idea can only signify something of which an image can be formed in the mind. We may have an idea or image of a mountain, a tree, a building; but we cannot surely have an idea or image of an argument or proposition.

— Boswell’s Life of Johnson

The Singapore Bar Model is a neat bit of maths pedagogy that has great potential in Science education. Ben Rogers wrote an excellent post about it here. Contrary to Samuel Johnson’s view, the Bar Model does attempt to present an argument or proposition as an image; and in my opinion, does so in a way that really advances students’ understanding.

Background

The Bar Model was developed in Singapore in the 1980s and is the middle step in the intensely-focused concrete-pictorial-abstract progression model that many hold instrumental in catapulting Singapore to the top of the TIMSS and PISA mathematical rankings.

Essentially, the Bar Model attempts to use pictorial representations as a stepping-stone between concrete and abstract mathematical reasoning. The aim is that the cognitive processes encouraged by the pictorial Bar Models are congruent with (or at least, have some similarities to) the cognitive processes needed when students move on to abstract mathematical reasoning.

Applying the Bar Model to a GCSE lesson on Magnification

I was using the standard I-AM formula triangle with some GCSE students who were, frankly, struggling.

Although most science teachers use formula triangles, they are increasingly recognised as being problematic. Formula triangles are a cognitive dead end because they are a replacement for algebra, rather than a stepping stone that models more advanced algebraic manipulations.

Having recently read about the Bar Model, I decided to try to present the magnification problem pictorially.

“The actual size is 0.1 mm and the image size is 0.5 mm. What is the magnification?” was shown as:

From this diagram, students were able to state that the magnification was x 5 without using the formula triangle (and without recourse to a calculator!)

The above question was presented as:

Note that the 1:1 correspondence between the number of boxes and the amount of magnification no longer applies. However, students were still able to intuitively grasp that 100/0.008 would give the magnification of x12500 — although they did need a calculator for this one. (Confession: so did I!)

More impressively, questions such as “The actual length of a cell structure is 3 micrometres. The magnification is 1500. Calculate the image size” could be answered correctly when presented in the Bar Model format like this:

Students could correctly calculate the image size as 4500 micrometres without recourse to the dreaded I-AM formula triangle. Sadly however, the conversion of micrometres to millimetres still defeated them.

But this led me to think: could the Bar Model be adapted to aid students in unit conversion? I’m sure it could, but I haven’t thought that one through yet…

However, I hope other teachers apply the Bar Model to magnification problems and let me know if it does help students as much as I think it does.

## Corinne’s Shibboleth and Embodied Cognition

You can watch a bird fly by and not even hear the stuff gurgling in its stomach. How can you be so dead?

— R. A. Lafferty, Through Other Eyes

In modern usage, a shibboleth is a custom, tradition or speech pattern that can be used to distinguish one group of people from another.

The literal meaning of the original Hebrew word shibbólet is an ear of corn. However, in about 1200 BCE, the word was used by the victorious Gileadites to identify the defeated Ephraimites as they attempted to cross the river Jordan. The Ephraimites could not pronounce the “sh” sound and thus said “sibboleth” instead of “shibboleth”.

As the King James Bible puts it:

And the Gileadites took the passages of Jordan before the Ephraimites: and it was so, that when those Ephraimites which were escaped said, Let me go over; that the men of Gilead said unto him, Art thou an Ephraimite? If he said, Nay; Then said they unto him, Say now Shibboleth: and he said Sibboleth: for he could not frame to pronounce it right.

Judges 12:5-6

The same story is featured in the irresistible (but slightly weird) Brick Testament through the more prosaic medium of Lego:

Sadly, the story did not end well for the Ephraimites:

Then they took him, and slew him at the passages of Jordan: and there fell at that time of the Ephraimites forty and two thousand.

This leads us to Corinne’s Shibboleth: a question which, according to Dray and Manogoue 2002, can help us separate physicists from mathematicians, but with fewer deleterious effects for both parties than the original shibboleth.
Corinne’s Shibboleth

This is because (according to Dray and Manogoue) mathematicians “view functions as maps, taking a given input to a prescribed output. The symbols are just placeholders, with no significance.” However, physicists “view functions as physical quantities. T is the temperature here; it’s a function of location, not of any arbitrary labels used to describe the location.”

Redish and Kuo 2015 comment further on this

[P]hysicists tend to answer that T(r,θ)=kr2 because they interpret x2+ y2 physically as the square of the distance from the origin. If r and θ are the polar coordinates corresponding to the rectangular coordinates x and y, the physicists’ answer yields the same value for the temperature at the same physical point in both representations. In other words, physicists assign meaning to the variables x, y, r, and θ — the geometry of the physical situation relating the variables to one another.

Mathematicians, on the other hand, may regard x, y, r, and θ as dummy variables denoting two arbitrary independent variables. The variables (r, θ) or (x, y) do not have any meaning constraining their relationship.

I agree with the argument put forward by Redish and Kuo that the foundation for understanding Physics is embodied cognition; in other words, that meaning is grounded in our physical experience.

Equations are not always enough. To use R. A Lafferty’s picturesque phraseology, ideally physicists should be able to hear “the stuff gurgling” in the stomach of the universe as it flies by….

Dray, T. & Manogoue, C. (2002). Vector calculus bridge project website, http://www.math.oregonstate.edu/bridge/ideas/functions

Redish, E. F., & Kuo, E. (2015). Language of physics, language of math: Disciplinary culture and dynamic epistemology. Science & Education, 24(5-6), 561-590.

## The p-prim path to enlightenment…?

The Duke of Wellington was once asked how he defeated Napoleon. He replied: “Napoleon’s plans were made of wire. Mine were made of little bits of string.”

In other words, Napoleon crafted his plans so thay they had a steely, sinewy strength that carried them to completion. Wellington conceded that his plans were more ramshackle, hand-to-mouth affairs. The difference was that if one of of Napoleon’s schemes broke or miscarried, it proved impossible to repair. When Wellington’s plans went awry, he would merely knot two loose bits of string together and carry on regardless.

I believe Andrea diSessa (1988) would argue that much of our knowledge, certainly emergent knowledge, is in the form of “little bits of string” rather than being organised efficiently into grand, coherent schemas.

For example, every human being has a set of conceptions about how the material world works that can be called intuitive physics. If a ball is thrown up in the air, most people can make an accurate prediction about what happens next. But what is the best description of the way in which intuitive physics is organised?

diSessa identifies two possibilities:

The first is an example of what I call “theory theories” and holds that it is productive to think of spontaneously acquired knowledge about the physical world as a theory of roughly the same quality, though differing in content from Newtonian or other theories of the mechanical world [ . . .]

My own view is that . . . intuitive physics is a fragmented collection of ideas, loosely connected and reinforcing, having none of the commitment or systematicity that one attributes to theories.

[p.50]

diSessa calls these fragmented ideas phenomenological primitives, or p-prims for short.

David Hammer (1996) expands on diSessa’s ideas by considering how students explain the Earth’s seasons.

Many students wrongly assume that the Earth is closer to the Sun during summer. Hammer argues that they are relying, not on a misconception about how the elliptical nature of the Earth’s orbit affects the seasons, but rather on a p-prim that closer = stronger.

The p-prims perspective does not attribute a knowledge structure concerning closeness of the earth and sun; it attributes a knowledge structure concerning proximity and intensity, Moreover, the p-prim closer means stronger is not incorrect.

[p.103]

diSessa and Hammer both argue that a misconceptions perspective assumes the existence of a stable cognitive structure where, in fact, there is none. Students may not have thought about the issue previously, and are in the process of framing thoughts and concepts in response to a question or problem. In short, p-prims may well be a better description of evanescent, emergent knowledge.

Hammer points out that the difference between the two perspectives has practical relevance to instruction. Closer means stronger is a p-prim that is correct in a wide range of contexts and is not one we should wish to eliminate.

The art of teaching therefore becomes one of refining rather than replacing students’ ideas. We need to work with students’ existing ideas and knowledge — piecemeal, inarticulate and applied-in-the-wrong-context as they may be.

Let’s get busy with those little bits of conceptual string. After all, what else have we got to work with?

REFERENCES

diSessa, A. (1988). “Knowledge in Pieces”. In Forman, G. and Pufall, P., eds, Constructivism in the Computer Age, New Jersey: Lawrence Erlbaum Publishers

Hammer, D. (1996). “Misconceptions or p-prims” J. Learn Sci 5 97

## Learning Is For The Birds

​Well versed in the expanses
that stretch from earth to stars,
we get lost in the space
from earth up to our skull.

Wislawa Szymborska, To My Friends

What do we mean by learning? To tell the truth, even as a teacher of twenty-five years experience, I am not sure.

Professor Robert Coe has suggested that learning happens when people have to think hard. In a similar vein, Daniel Willingham contends that knowledge is the residue of thought. Siegfried Engelmann proposes that learning is the capacity to generalise to new examples from previous examples. I have also heard learning defined as a change in the long term memory.

One thing is certain, learning involves some sort of change in the learner’s brain. But what is acknowledged less often is that it doesn’t just happen in human brains.

Contrary to standard social science assumptions, learning is not some pinnacle of evolution attained only recently by humans. All but the simplest animals learn . . . [And some animals execute] complicated sequences of arithmetic, logic, and data storage and retrieval.
— Steven Pinker, How The Mind Works (1997), p.184

An example recounted by Pinker is that of some species of migratory birds that fly thousands of miles at night and use the constellations to find North. Humans do this too when we find the Pole Star.

But with birds it’s surely just instinct, right?

Wrong. This knowledge cannot be genetically “hardwired” into birds as it would soon become obsolete. Currently, a star known as Polaris just happens to be (nearly) directly above the Earth’s North Pole, so that as the Earth rotates on its axis, this star appears to stand still in the sky while the other stars travel on slow circular paths. But it was not always thus.

The Earth’s axis wobbles slowly over a period of twenty six thousand years. This effect is called the precession of the equinoxes. The North Star will change over time, and oftentimes there won’t be star bright enough to see with the naked eye at the North Celestial Pole for there to be “North Star” — just as currently there is no “South Star”.But there will be one in the future, at least temporarily, as the South Celestial Pole describes its slow precessional dance.

Over evolutionary time, a genetically hardwired instinct that pointed birds towards any current North Star or South Star would soon lead them astray in a mere few thousand years or so.

So what do the birds do?

[T]he birds have responded by evolving a special algorithm for learning where the celestial pole is in the night sky. It all happens while they are still in the nest and cannot fly. The nestlings gaze up at the night sky for hours, watching the slow rotation of the constellations. They find the point around which the stars appear to move, and record its position with respect to several nearby constellations. [p.186]

And so there we have it: the ability to learn confers an evolutionary advantage, amongst many others.