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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…

If you find these Excel workbooks useful, please leave a comment on this blog or Tweet a link (please add @emc2andallthat to alert me).

Happy graphing, folks 🙂

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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.

Screenshot 2020-03-22 at 14.39.37.png

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.

Screenshot 2020-03-22 at 16.08.06.png

Two conductors viewed from the inertial frame of the laboratory

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.

Screenshot 2020-03-22 at 16.13.28.png

The same two conductors viewed from the inertial frame of one of the positive charges in conductor A. Note that all the positive charges are now stationary; the laboratory is moving with speed v right to left, and the negative charges are moving with speed 2v right to left

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.

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It is F=ma, you know

“It is cheese. (Caerphilly.)”

Main research finding of the first manned Welsh mission to the Moon, as reported by Max Boyce c. 1974.

A Brief History of F=ma

When writing A Brief History Of Time, it is said that a literary agent warned Stephen Hawking that each mathematical equation that he included in the final draft would halve its eventual sales.

And one of the complex equations that Hawking wished to include? A summary of Newton’s 2nd Law: F=ma.

In other words, the force F acting on an object is equal to the mass m of the object multiplied by the acceleration experienced by the object.

(In the end, Hawking opted to include only Einstein’s E=mc2 in what turned out to be the ultra bestselling A Brief History of Time.)


F=ma is Newton’ s 2nd Law: NOT!!!

@SciByDegrees wrote an interesting post arguing that the old Physics teacher’s standby of summarising Newton’s Second Law of Motion (N2) as F=ma is wrong.

The gist of his argument (and it’s a hard argument to counter) is that F=d(mv)/dt is a far better expression of the law than the F=ma version because it covers a wider range of circumstances.

This states N2 in terms of momentum, where momentum is the product of mass m multiplied by the velocity v. More exactly, it says that force acting on an object is equal to the object’s rate of change of momentum: or, if you prefer, the change in momentum divided by the time taken for the change is equal to the force.

This is the version of N2 stated in most dictionaries of Physics. For example, the Oxford Dictionary of Physics (2015) p. 383.

I know, because on reading @SciByDegrees’ post I immediately looked up N2 in the Dictionary with the express intention of countering the argument. Imagine my consternation and horror when I found that I was wrong. (Actually, not that much consternation and horror: I am fairly inured to being wrong as it happens fairly often…)

The argument suggests that, just like V=IR is not a statement of Ohm’s Law unless R has a fixed value (like a fixed length of wire at a constant temperature), F=ma is not a sufficient statement of N2 unless the mass m is constant.

For example, if we consider a rocket capable of producing a steady 1000 N of thrust; at t=0 its mass is (say) 10 kg so its acceleration is 100 m/s2. However at t=1 s its mass has decreased by 1 kg so the acceleration is now 111 m/s2 even though the thrust is still 1000 N so obviously F is not proportional to m so F does not equal ma in this situation.


Feynman as the new Aristotle

Richard Feynman (1965) wrote along similar lines in his justly famous Lectures on Physics:

Thus at the beginning we take several things for granted. First, that the mass of an object is constant; it isn’t really, but we shall start out with the Newtonian approximation that mass is constant, the same all the time, and that, further, when we put two objects together, their masses add. These ideas were of course implied by Newton when he wrote his equation, for otherwise it is meaningless. For example, suppose the mass varied inversely as the velocity; then the momentum would never change in any circumstance, so the law means nothing unless you know how the mass changes with velocity. At first we say, it does not change.

However, I think Feynman is considerably oversimplifying what Newton said here. Dare one suppose that Feynman, who had an enviable natural facility for talking intelligently and arrestingly about nearly any subject under the Sun, had perhaps skimped a little on his background reading?

Incidentally, does anyone think that physicists (especially physics educators — myself included) are beginning to treat Feynman as the medieval scholastics are reputed to have treated Aristotle? That is to say, he is regarded as the final word on everything; or, at least, everything physics-related in the case of Feynman.


What Would Newton Do?

George Smith (2008) points out that:

The modern F=ma form of Newton’s second law nowhere occurs in any edition of the Principia [ . . . ] Instead, it has the following formulation in all three editions: A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed. In the body of the Principia this law is applied both to discrete cases, in which an instantaneous impulse such as from impact is effecting the change in motion, and to continuously acting cases, such as the change in motion in the continuous deceleration of a body moving in a resisting medium. Newton thus appears to have intended his second law to be neutral between discrete forces (that is, what we now call impulses) and continuous forces.

This, I think, supports my contention that F=ma is as good a modern reformulation of Newton’s 2nd Law as any other.

If we go back to the rocket example, the instantaneous acceleration at t=0 and t=1 s can be calculated using F=ma (provided we take account of the change in m, of course). In effect, we are considering the change in motion due an instantaneous impulse here.

Please note that I would cheerfully concede that F=d(mv)/dt would yield a better and more productive analysis of rocket motion if we are considering the continuous action of the force over time rather than at isolated instants.

The analogy with V=IR is useful here. V is always equal to I times R but V is only directly proportional to I over a continuous range of values of I for a limited set of conductors we call Ohmic conductors whose resistance R is fixed over a range of physical conditions. Likewise, F is always equal to m times a but F is only directly proportional to a for a continuous range of values of a when we are considering a system whose mass is fixed.

As V=IR is neutral with respect to whether R is fixed is not, I believe that F=ma is neutral with respect to whether m is fixed or not.


Will the Real Second Law Please Stand Up?

What is Newton’s Second Law? Is it a definition of force? Is it a definition of mass? Or is it an empirical proposition linking force, mass and acceleration?

Brian Ellis (1965) argues that it partakes of all three:

Consider how Newton’s second law is actually used. In some fields it is unquestionably true that Newton’s second law is used to define a scale of force. How else, for example, can we measure interplanetary gravitational forces? But it is also unquestionably true that Newton’s second law is sometimes used to define a scale of mass. Consider, for example, the use of mass spectrography. And in yet other fields, where force, mass and acceleration are all easily and independently measurable, Newton’s second law of motion functions as an empirical correlation between these three quantities. Consider, for example, the application of Newton’s second law in ballistics and rocketry [ . . .] To suppose that Newton’s second law of motion, or any law for that matter, must have a unique role that we can describe generally and call the logical status is an unfounded and unjustifiable supposition.

In some senses, I suppose we might like those unfortunate nations in Gulliver’s Travels who fought a long and bitter war over the question of whether one should eat a boiled egg from the pointed or rounded end:

During the course of these troubles, the emperors of Blefusca … accusing us of making a schism in religion, by offending against a fundamental doctrine of our great prophet Lustrog, in the fifty-fourth chapter of the Blundecral (which is their Alcoran). This, however, is thought to be a mere strain upon the text; for the words are these: ‘that all true believers break their eggs at the convenient end.’ And which is the convenient end, seems, in my humble opinion to be left to every man’s conscience.


REFERENCES

Ellis, Brian. “The origin and nature of Newton’s laws of motion.” Beyond the edge of certainty (1965): 29-68.

Feynman, R. P., Leighton, R. B., & Sands, M. (1965). The Feynman Lectures on Physics; vol. 1 (Accessed from http://www.feynmanlectures.caltech.edu/I_09.html on 11/4/19)

Smith, George, “Newton’s Philosophiae Naturalis Principia Mathematica“, The Stanford Encyclopedia of Philosophy (Winter 2008 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/win2008/entries/newton-principia/&gt;.ewto

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Dual-coding SUVAT Problems

The theory of dual coding holds that the formation of mental images, in tandem with verbal processing, is often very helpful for learners. In other words, if we support verbal reasoning with visual representations, then better learning happens.

Many years ago, I was taught the dual coding technique outlined below to help with SUVAT problems. Of course, it wasn’t referred to as “dual coding” back then, but dual coding it most definitely is.

I found it a very useful technique at the time and I still find it useful to this day. And what is more, it is in my opinion a pedagogically powerful procedure. I genuinely believe that this technique helps students understand the complexities and nuances of SUVAT because it brings many things which are usually implicit out into the open and makes them explicit.

SUVAT: “Made darker by definition”?

BOSWELL. ‘He says plain things in a formal and abstract way, to be sure: but his method is good: for to have clear notions upon any subject, we must have recourse to analytick arrangement.’

JOHNSON. ‘Sir, it is what every body does, whether they will or no. But sometimes things may be made darker by definition. I see a cow, I define her, Animal quadrupes ruminans cornutum. But a goat ruminates, and a cow may have no horns. Cow is plainer.

— Boswell’s Life of Johnson (1791)

As I see it, the enduring difficulty with SUVAT problems is that such things can indeed be made darker by definition. Students are usually more than willing to accept the formal definitions of s, u, v, a and t and can apply them to straightforward and predictable problems. However, the robotic death-by-algorithm approach fails all too frequently when faced with even minor variations on a theme.

Worse still, students often treat acceleration, displacement and velocity as nearly-synonymous interchangeable quantities: they are all lumped together in that naive “intuitive physics” category called MOVEMENT.

The approach that follows attempts to make students plainly see differences between the SUVAT quantities and, hopefully, as make them as plain as a cow (to borrow Dr Johnson’s colourful phrasing).

Visual Symbols for the Dual-coding of SUVAT problems

Screenshot 2018-12-25 at 12.02.38.png

1.1 Analysing a simple SUVAT problem using dual coding

Problem: a motorcycle accelerates from rest at 0.8 m/s2 for a time of 6.0 seconds. Calculate (a) the distance travelled; and (b) the final velocity.

Screenshot 2018-12-25 at 12.09.42.png

Please note:

  1. We are using the AQA-friendly convention of substituting values before rearrangement. (Some AQA mark schemes award a mark for the correct substitution of values into an expression; however, the mark will not be awarded if the expression is incorrectly rearranged. Weaker students are strongly encouraged to substitute before rearrangement, and this is what I model.)
  2. A later time is indicated by the movement of the hands on the clock.

So far, so blindingly obvious, some might say.

But I hope the following examples will indicate the versatility of the approach.

1.2a Analysing a more complex SUVAT problem using dual coding (Up is positive convention)

Problem: A coin is dropped from rest takes 0.84 s to fall a distance of 3.5 m so that it strikes the water at the bottom of a well. With what speed must it be thrown vertically so that it takes exactly 1.5 s to hit the surface of the water?

Screenshot 2018-12-25 at 14.33.25.png

Another advantage of this method is that it makes assigning positive and negative directions to the SUVAT vectors easy as it becomes a matter of simply comparing the directions of each vector quantity (that is to say, s, u, v and a) with the arbitrarily selected positive direction arrow when we substitute values into the expression.

But what would happen if we’d selected a different positive direction arrow?

1.2b Analysing a more complex SUVAT problem using dual coding (Down is positive convention)

Problem: A well is 3.5 m deep so that a coin dropped from rest takes 0.84 s to strike the surface of the water. With what speed must it be thrown so that it takes exactly 1.5 s to hit the surface of the water?

Screenshot 2018-12-25 at 14.43.42.png

The answer is, of course, numerically equal to the previous answer. However, following the arbitrarily selected down is positive convention, we have a negative answer.

1.3 Analysing a projectile problem using dual coding

Let’s look at this typical problem from AQA.

Screenshot 2018-12-25 at 14.50.12.png

We could annotate the diagram like this:

Screenshot 2019-01-03 at 18.30.09.png

Guiding our students through the calculation:

Screenshot 2019-01-03 at 18.34.19.png

Just Show ‘Em!

Some trad-inclined teachers have embraced the motto: Just tell ’em!

It’s a good motto, to which dual coding can add the welcome corollary: Just show ’em!

The Unreasonable Effectiveness of Mathematics in the Natural Sciences

The famous phrase is, of course, from physicist Eugene Wigner (1960: 2):

My principal aim is to illuminate it from several sides. The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it.

Further exploration of the above problem using dual coding can, I believe, give A-level students a glimpse of the truth of Wigner’s phrase.

This Is The Root You’re Looking For

In the calculation above, we found that when s = -1.8 m, v could have a value of plus or minus 6.90 m/s. Since we were interested in the velocity of the kite boarder at the end of the journey, we concluded that it was the negative root that was significant for our purposes.

But does the positive root have any physical significance? Why yes, it does. It indicates the other possible value of v when s = -1.8 m.

The displacement was -1.8 m at only one point on the real journey. However, if the kite boarder had started their projectile motion from the level of the water surface instead of from the top of the ramp, their vertical velocity at this point would have been +6.9 m/s.

Screenshot 2019-01-04 at 14.24.14.png

The fact that the kite boarder did not start their journey from this point is immaterial. Applying the mathematics not only tells us about their actual journey, but all other possible journeys that are consistent with the stated parameters and the subset of the laws of physics that we are considering in this problem — and that, to me, borders enough on the mysterious to bring home Wigner’s point.

And finally…

Screenshot 2019-01-04 at 15.11.47.png

This information allows us to annotate our final diagram as below (bearing in mind, of course, that the real journey of the kite boarder started from the top of the ramp and not from the water’s surface as shown).

Screenshot 2019-01-04 at 15.14.20.png

Let me end on a more cheerful note. The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.

Wigner 1960: 9

Reference

Wigner, E. (1960). The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Communications in Pure and Applied Mathematics; Vol. 13, No. 1.

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My Best Blogs of 2018

I am very pleased to say that I got my best annual viewing figure ever: just over 9000 views in total beating a previous best of 7000.

Small beer for some in the edu-blogosphere perhaps, but I am well chuffed.

And my most popular blogs were (in reverse order):

  • 5) IoP Energy for Busy Teachers. This is yet another of my periodic tilts at the Institute of Physics’ revised schema for teaching energy, including some attempted humour.
  • 4) The FBI and Gang Sign for Physicists. I am am at a loss to explain why this random stream-of-edu-conciousness post from 2016 seems to get a slow but steady stream of readers, mainly from the U.S.A. for some reason
  • 3) Teaching Electric Circuits? Climb On Board The Coulomb Train! This, I have to admit, is one of my personal favourites. Although a persuasive case can be made for the rope model (I’m looking at you, @PhysicsUK and this), the CTM (Coulomb Train Model) is still the best IMHO. For example, which one would be the best when we’re considering RMS values, huh?
  • 2) Two posts on applying the Singapore Bar Model to GCSE Science Topics and ditching those horrible, horrible formula triangles. The first was on Magnification and the second on Kinetic Energy.
  • 1) And my all-time most-viewed blog post is … MARKOPALYPSE NOW!!! (And thanks to Adam Boxer’s A Chemical Orthodoxy for a link that generated many of the views.)

Bubbling under, we have my contribution to the #CurriculumInScience symposium, Using P-Prims For Fun And Profit. I’m hoping this will get a few more views in the New Year.

And on that note: thanks for reading this far and Happy New Year everyone.

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Using P-prims For Fun And Profit

This is my contribution to the #CurriculuminScience symposium. You can read the first contribution from Ruth Walker here. The next contribution from Jasper Green can be found here.

“She said she was going to join a church as soon as she decided which one was right. She never did decide. She did develop a terrific hankering for a crucifix, though. And she bought one from a Santa Fe gift shop during a trip the little family made out West during the Great Depression. Like so many Americans, she was trying to construct a life that made sense from things she found in gift shops.

— Kurt Vonnegut, Slaughterhouse-Five [emphasis added]

Introduction

It was never supposed to be like this, of course. Many of the great thinkers of the past conceived of the human mind as a vast pyramid: either an inverted pyramid resting on an apex consisting of a single, unfalsifiable thought such as “I think therefore I am” as Rationalists such as Descartes posited; or, alternatively, as a pyramid resting on a base of simple sense-impressions as Empiricists such as Locke suggested.

pyramid

The examples of entities that exist are taken from Douglas Adams’ The Hitchhiker’s Guide To The Galaxy, where the sentient computer Deep Thought started with “I think therefore I am” and deduced the existence of rice pudding and income tax before anyone managed to switch it off.

The truths emerging from modern cognitive science indicate that things are a good deal more complicated and messier than either the Rationalists or Empiricists supposed.

In fact, all of us are closer to Mrs Pilgrim in Vonnegut’s Slaughterhouse-Five than we would generally like to admit: the uncomfortable truth is that we are all closer to opportunistic concept-grubbing, “gift shop”-magpies than the systematic pyramid-masons of either Rationalist or Empiricist thought. Each and every one of us is, to a greater or lesser degree, “trying to construct a life that makes sense” from random things that we find lying around in real or metaphorical gift shops.

Perhaps (of all people!) Dashiell Hammett put it best:

“Nobody thinks clearly, no matter what they pretend. Thinking’s a dizzy business, a matter of catching as many of those foggy glimpses as you can and fitting them together the best you can. That’s why people hang on so tight to their beliefs and opinions; because, compared to the haphazard way in which they arrived at, even the goofiest opinion seems wonderfully clear, sane, and self-evident. And if you let it get away from you, then you’ve got to dive back into that foggy muddle to wangle yourself out another to take its place.”
— Dashiell Hammett, The Dain Curse

Defeat From The Jaws of Victory: “Here’s to you, Mrs Pilgrim.”

Andrea diSessa (1996) recounts a series of interviews with “J”, a freshman undergraduate student of Physics at university. During one interview, J was asked to explain the physics of throwing a ball up into the air. She recounted a near-perfect, professional physicist-level analysis of the phenomenon, noting (correctly) that after the ball leaves the hand the “only force acting on it is gravity”. However, when diSessa asked the seemingly innocuous question about what happens at the peak of the toss:

Rather than produce a straightforward answer, J proceeded to reformulate her description of the toss. The reformulation is not instantaneous . . . Strikingly she winds up with an “impetus theory” account of the toss. “Your hand imparts a force that at first overcomes gravity, but gradually dies away. At the peak, there a balance of forces, which is broken as the internal force fades further and gravity takes over.”

In other words, even a student of Physics, educated to a much higher level of domain-specific knowledge than the typical layperson, can be persuaded to retreat back into the ”foggy muddle” with surprising ease. In other words, even the very best of us can snatch defeat from the jaws of victory all too easily.

diSessa (1988) explains this and similar models as part of the KIP model (Knowledge in Pieces). For example:

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

The basic “atom” or building block of this empirical model is the p-prim or phenomenological primitive.

P-prims are elements of intuitive knowledge that constitute people’s “sense of mechanism”, their sense of which happenings are obvious, which are plausible, which are implausible, and how one can explain or refute real or imagined possibilities. [diSessa 2018: 69]

P-prims are abstractions of familiar events that come to serve as explanations as they are applied to a wider range of contexts. The p indicates that they originate from the phenomenologically-rich and lived experience of human beings; the prim indicates that they are primitive in the sense that they sufficient explanations of phenomena. Once a p-prim is invoked, usually no further explanation is required or possible: “That’s just the way it is.” Examples of p-prims suggested by diSessa [1996: 716] are:

  • The “Ohm’s Law” p-prim: the idea that an outcome increases as a “force” increases, but decreases as the “resistance” increases.
  • The “Balance and Equilibrium” p-prim: systems which are “in balance” will be stable; systems which are “out of balance” will naturally and spontaneously return to equilibrium.
  • The “Blocking and Guiding” p-prim: solid and stable objects can stop objects moving without applying a force; tubes and railway tracks can also “guide” moving objects without applying any force.
  • The “Dying Away” p-prim: lack of motion or activity is the natural state of inanimate objects; if disturbed, they will naturally return to this state as the perturbation “dies away’

P-prims are subconceptual: they comprise a fluid and changeable layer below concepts and beliefs. Humans may have hundreds if not thousands of p-prims. There is no strict hierarchy: we may shift from one p-prim to another with simply a shift of attention. Where multiple p-prims conflict, one facet of the situation may cue the application of a particular p-prim rather than another. [see diSessa 1996: 715]

The Wrath of Kuhn: “So You Say You Want a Revolution?”

In his hugely-influential The Structure of Scientific Revolutions (1970), Thomas Kuhn suggested that scientific progress had two distinct phases:

  • Normal Science, where essentially scientists engaged in puzzle-solving activity but where the guiding paradigm or disciplinary matrix of the science is more or less accepted without question. An example might be pre-Copernican astronomy where astronomers made observations and predictions without questioning the geocentric model of the Solar System;
  • Revolutionary Science, where scientists realise their previously-successful paradigm is no longer able to adequately explain observed phenomena. An example might be the rejection of the Newtonian paradigm and the acceptance of Einsteinian relativistic physics in the early 1900s.

Scientific progress was thus viewed as a gestalt switch between two incommensurable systems of knowledge. One either sees a “Newtonian”-duck, or a “Relativistic”-rabbit. One cannot see both simultaneously.

gestalt.png

One can either see a duck or a rabbit: but not both at the same time.

Kuhn’s work was immensely influential (perhaps overly influential) in a number of spheres; in the context of education, the heady seductiveness of Kuhn’s approach directly influenced what diSessa [2014: 5] dubs the “misconceptions movement”.

Broadly speaking, proponents thought that students had deeply entrenched but false beliefs. The solution seemed obvious: these false beliefs were barriers to learning that had to be rooted out and overcome (c.f. the Ohm’s Law p-prim above!) . Students had to be persuaded to ditch their false beliefs and accept the correct ones.

But what was the nature of these false beliefs? diSessa [2014:7] argues that some like Carey (1985) drew explicit parallels with Kuhn’s work, arguing that children undergo a paradigm-shift at about 10-years-old when they recognise that inanimate objects do not have intentions and begin to think of “alive” as describing a set of mechanistic processes. Others (argues diSessa) like McCloskey (1983) supposed that students begin school physics with a well-formed, coherent and articulate theory (with parallels to early medieval scientists such as Buridan and Galileo) that directly competes with and interferes with their acceptance of Newtonian physics.

However, all of these approaches can be categorised as being part of the “Misconceptions Movement”.

Yin vs. Yang: Positive and Negative Influences of the Misconceptions Movement

A positive influence of misconceptions studies was bringing the importance of educational research into practical instructional circles. Teachers saw vivid examples of students responding to apparently simple conceptual questions in incorrect ways. Poor performance in response to basic questions, often years into instruction, could not be dismissed.

[diSessa 2014: 6]

Another hugely positive influence of Misconceptions research was that it showed that students were not “blank slates” and that prior knowledge had a strong influence on future learning.

However, according to diSessa the misconceptions movement also had some pernicious negative influences:

  • It emphasised the negative contributions of prior knowledge: it almost exclusively characterised prior knowledge as either false or unhelpful which led to “conflict” models of instruction. Ironically, the explicit detailing of “wrong” ideas in order to “overcome” them led to them being strengthened for some students.
  • How learning was possible was not a matter that was often discussed in detail. The depth, coherence or strength of particular misconceptions was not always assessed: were they simply isolated beliefs or coherent theories of a similar nature to those held by working scientists? As a result, practical guidance on how to teach particular concepts was not always forthcoming.

Tourist: “Is This Way To Amarillo?” Local: “Well, I wouldn’t start from here if I were you.”

As a working Physics teacher, one of the most useful teaching tools that I’ve begun using as a result of becoming aware of diSessa’s work, is that of a bridging analogy. This approach was outlined by Hammer 2000: S54-55. For example, how can we successfully introduce the idea of a normal reaction force, say in the context of a book resting on the surface of a table?

Students often invoke the “blocking” p-prim in this context. The table passively “blocks” the action of gravity — and that’s all there is to it.

However, a bridging analogy can be used here. Show an object resting on (and compressing) a spring; identify the forces acting on the object. Because the spring is an “active” component in this situation, students can accept that pushing down on it produces an upward “reaction force”. One can then extend this to (say) a student sitting on a plank (which “bows” slightly with their weight) and then apply it to more stable structure such as a table which exhibits no visible “bowing”.

I have found such approaches to be the most productive: in other words, we aim to work around the p-prim rather than attacking the p-prim head on, and along the way we try to get our students to activate more helpful p-prims that have more direct applicability to the context.

As teachers, we only very rarely have the luxury of choosing our students’ starting points. There is no “Well, if you want to get where you’re going, I wouldn’t start from here if I were you.”

We are teachers. Whatever the situation, we start from where our students start. Ladies and gentlemen, we start from here.

References

Carey, S. (1985). Conceptual change in childhood. Cambridge, MA: MIT Press/Bradford Books

diSessa, A. A. (1988). Knowledge in pieces. In G. Forman & P. B. Pufall (Eds.), Constructivism in the computer age (pp. 49-70). Hillsdale, NJ, US: Lawrence Erlbaum Associates, Inc.

diSessa, A. A. (1996). What do” just plain folk” know about physics. The handbook of education and human development: New models of learning, teaching, and schooling, 709-730. [Accessed from http://www.staff.science.uu.nl/~savel101/fi-msecint/literature/disessa1996.pdf on 22/10/18]

DiSessa, A. A. (2014). A history of conceptual change research: Threads and fault lines. [Accessed from https://escholarship.org/uc/item/1271w50q on 22/10/18]

diSessa, A. A. (2018). A Friendly Introduction to “Knowledge in Pieces”: Modeling Types of Knowledge and Their Roles in Learning. In Invited Lectures from the 13th International Congress on Mathematical Education (pp. 65-84). Springer International Publishing. [Accessed from https://link.springer.com/chapter/10.1007/978-3-319-72170-5_5 on 22/10/18]

Hammer, D. (2000). Student resources for learning introductory physics. American Journal of Physics, 68(S1), S52-S59 [Accessed from http://oap.nmsu.edu/JiTT_NMSU_workshop/pdfs/StudentResourcesHammer.pdf on 22/10/18]

McCloskey, M. (1983). Naive theories of motion. In D. Gentner and A. Stevens (Eds.) Mental Models (pp. 299-323). Hillsdale, NJ: Lawrence Erlbaum Associates.

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Ohm From Ohm

Amongst the myriad inconveniences and troubles of a Physics teacher’s life, the choice of the symbols commonly used to represent voltage, current and resistance, must surely rank in the top ten.

V is for voltage in volts, V

Well, OK, that’s sensible enough. On a good day, I may even remember to call it “potential difference”. The sage advice of Never use two words when one will do is commonly accepted by everyone; however, Physics teachers have, as a profession, decided to go it alone and completely ignore this tired old saw. Thus, voltage is become potential difference because of — erm, reasons (?)

One can only hope that everyone got the memo . . .

R is for resistance in ohms, Ω

R for resistance? That’s fairly sensible too.

“But what’s that weird squiggly thing, Sir?”

“Ah, you mean the Greek letter omega? Because Physics is soooo enormous that the measly 26 letters of the Latin alphabet ain’t big enough for it…”

I is for current in amps, A

“WTφ? Are you taking the πΣΣ, Sir?”

“I know, I know! Look, if it helps, think of it as short for intensité du courant . . . Wait, don’t leave! Stop, I have many more fun Physics facts to teach you! Look, here’s a picture of Richard Feynman playing his bongo drums — nooooooooo!”

Ohm’s Law: or is it more a sort of guideline?

Let’s start with a brief statement of Ohm’s Law:

V = I R

Except, that’s not Ohm’s Law; it’s actually the definition of resistance:

R = V / I

There is not a single instance where it is not true by definition. The value of resistance will always be equal to the ratio of the potential difference and the current.

Think of it like this. At room temperature, 1 V of potential difference can push (say) 0.5 A of current through the wire in a filament bulb. (I just love that retro 1890s tech, don’t you?)

This means it has a resistance of 1/0.5 = 2 ohms. However, bump up the potential difference to 6 V and the current is (say) 0.75 A. This means that is has a resistance of 6/0.75 = 8 ohms. Its resistance has changed because it has become hotter. In other words, its resistance is not constant.

Ohm’s Law is perhaps most simply stated as:

The potential difference is directly proportional to the current over a range of physical conditions (including temperature).

Using standard symbols:

V α I

or, taking R’ as a constant of proportionality:

V = I R’

You do see the difference, don’t you?

In the first example, R is not a constant value for a given range of physical conditions: for example it can get higher as the temperature increases.

In the second, R’ is constant over a range of temperatures and other physical conditions.

And so there we have it: V=IR can be a perfectly valid statement of Ohm’s Law, provided it is specified that R is constant. If one does not do that, then all bets are off…

In the meantime, here’s another picture of Richard Feynman playing the bongo drums. Enjoy!

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Lottie and Lorentzian Length Contraction

@_youhadonejob tweeted this textbook picture with the amusing and sardonic comment “Little girl in this textbook is 5 m tall”.

I liked @jim_henderson60’s take on this when he tweeted: “You see. Physics helps us all grow tall.”

But then I started thinking, what if the 5 m measuring stick was in an inertial frame moving past Lottie’s inertial frame at a substantial fraction of light speed? (In my head, I named the girl “Lottie”, although “Alice” would be more in the more usual tradition of SR* pedagogy, I guess.)

The illustration could represent that single instant at which both ends of the 5 m ruler were precisely opposite Lottie’s head and feet as its inertial frame whizzed by hers…

A quick calculation indicated that Lorentz length contraction could indeed account for the relative measurements on the illustration if v = 0.97c

Of course, Lorentz length contraction is a two way street. From the 5 m ruler’s inertial frame, length contraction would make Lottie appear even shorter than her compact 1.2 m. Given that v = 0.97c, I calculate that she would appear only 0.29 m tall.

Correction: not appear. She would genuinely be only 0.29 m tall when viewed from that inertial frame, just as the 5 m rule would genuinely be only 1.2 m long when viewed from Lottie’s inertial frame.

We live in an universe where everything is indeed relative. However, for most of us that takes a fair amount of getting used to…

*SR = special relativity. My brain is currently too small to handle GR (general relativity).

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Markopalypse Now

AHT VAL: And once you’ve finished marking your students’ books and they have responded IN DETAIL to your DETAILED comments, you must take them in again and mark them a second time using a different coloured pen!

AHT HARVEY: A page that’s marked in only one colour is a useless page!

NQT BENJAMIN: Erm, if you say so. But why?

AHT VAL: It’s basic Ofsted-readiness, Benjamin. Without a clearly colour-coded dialogue between teacher and student, how can we prove that the student has made progress as a result of teacher feedback?

NQT BENJAMIN: But I’ve only got this red biro…

AHT HARVEY GRINS UNPLEASANTLY AND OPENS A CABINET FULL OF PENS OF MANY COLOURS.

AHT HARVEY: In this school we wage a constant battle against teacher sloth and indifference!

(With apologies to The League Of Gentlemen)

I have been a teacher for more than 26 years and I tell you this: I have never marked as much or as often as I am now. We are in the throes of a Marking Apocalypse — a Markopalypse, if you will.

And why am I doing this? Have I had a Damascene-road conversion to the joy of rigorous triple marking?

No. I do it because I have to. I do it because of my school’s marking policy. More to the point, I do it because my school expends a great deal of time and energy checking that their staff is following the policy. And my school is not unique in this.

Actually, to be fair, I think my current school has the most nearly-sensible policy of the three schools I have worked in most recently, but it is still an onerous burden even for an experienced teacher who can take a number of time-saving short cuts in terms of lesson planning and preparation.

Many schools now include so-called “deep marking” or “triple marking” in their lists of “non-negotiables”, but there are at least two things that I think all teachers should know about these policies.

1. “We have to do deep/triple marking because of Ofsted”

No, actually you don’t. In 2016, Sean Harford (Ofsted National Director, Education) wrote:

[I]nspectors should not report on marking practice, or make judgements on it, other than whether it follows the school’s assessment policy. Inspectors will also not seek to attribute the degree of progress that pupils have made to marking that they might consider to be either effective or ineffective. Finally, inspectors will not make recommendations for improvement that involve marking, other than when the school’s marking/assessment policy is not being followed by a substantial proportion of teachers; this will then be an issue for the leadership and management to resolve.

2. “Students benefit from regular feedback”

Why yes, of course they do. But “feedback” does not necessarily equate to marking.

Hattie and Timperley write:

[F]eedback is conceptualized as information provided by an agent (e.g., teacher, peer, book, parent, self, experience) regarding aspects of one’s performance or understanding. A teacher or parent can provide corrective information, a peer can provide an alternative strategy, a book can provide information to clarify ideas, a parent can provide encouragement, and a learner can look up the answer to evaluate the correctness of a response. Feedback thus is a “consequence” of performance.

So a textbook, mark scheme or model answer can provide feedback. It does not have to be a paragraph written by the teacher and individualised for each student.

Daisy Christodoulo makes what I think is a telling point about the “typical” feedback paragraphs encouraged by many school policies:

[T]eachers end up writing out whole paragraphs at the end of a pupils’ piece of work: ‘Well done: you’ve displayed an emerging knowledge of the past, but in order to improve, you need to develop your knowledge of the past.’ These kind of comments are not very useful as feedback because whilst they may be accurate, they are not helpful. How is a pupil supposed to respond to such feedback? As Dylan Wiliam says, feedback like this is like telling an unsuccessful comedian that they need to be funnier.

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Whither Edu-blogging?

The task of an author is, either to teach what is not known, or to recommend known truths by his manner of adorning them.

Samuel Johnson, The Rambler, 27 March 1750

I regret to say that, for me at least, blogging has become a habit that has been more honoured in the breach than in the observance. And, judging from a conversation or two on Twitter, I haven’t been alone. A number of edu-bloggers also seem to have hit a dry spell.

Some ask: what’s the point? What have we actually achieved? In a typical school, how many teachers actually read any edu-blogs? Outside the edu-Twitter bubble, has anyone ever changed anybody else’s mind, ever? Humans can generally change their location easily enough, but as Horace observed mordantly many years ago, “Who can change their mind?”

And yet. Reading blogs and engaging in Twitter conversations has changed at least one person’s mind: mine. And one of the most important things it taught me was: I was not alone.

I wasn’t alone in thinking that group work was over-emphasised as a panacea to the point of absurdity. I wasn’t alone in thinking that Learning Styles seemed a bit dodgy. I wasn’t alone in believing that a teacher should, on occasions, be an unapologetic sage-on-the-stage and not a permanently-muted guide-on-the-side.

And, in my opinion, a number of things have indeed changed for the better. Ofsted still has issues but it isn’t the educational Thought Police which brooked no dissent from the One True Path that it was a few years ago. A significant part of the credit for this should go to the edu-blogging pioneers who pointed out that a number of its policies had no clothes, and did this using evidence and reasoned argument rather than merely relying on a set of appeals-to-educational-authority as was the style at the time. I would single out @oldandrewuk, @tombennett71 and @daisychristo as being particularly influential in this regard, but there were many others.

I agree with @larrylemonmaths‘ comment that “When the stonemason hits the rock, the first 99 times, it seems like nothing is happening, then suddenly, on the 100th blow, the rock breaks apart. It’s important to keep blogging and talking and arguing, even if it seems like nothing is happening.”

So if we are to continue blogging, what should we blog about? Whither Edu-blogging? in other words.

If I was to highlight some current issues that I think would benefit from more people blogging about them, they would be:

1. Markopalypse Now: why are most teachers in most schools marking so much? When did insane amounts of over-marking become the new normal? Do people realise that written marking is not the same as feedback and that the majority of marking is being undertaken to comply with school policy and a misguided idea of “what Ofsted wants”.

2. The Bonfire Of The Greyhairs: why are so many experienced teachers leaving the profession? Are some of them being forced out because of budgeting pressures with manufactured “performance issues”? Is there any other profession where the wisdom of long-serving colleagues is not only sidelined as an irrelevance but actively rejected?

3. Accountability Roulette And The Culture Of Fear: research suggests that the “teacher factor” is responsible for between 1 and 14% of educational outcomes. Why, then, are teachers judged as if they are accountable for 100%?

No doubt I will blog on other issues besides the ones above (assuming that I blog at all!), but I will try to contribute to the tap-tap-tap of stonemason’s chisels on the adamantine rock of these problems at least.

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