Resources Framework

Exploring J. S. Mill’s classification of misconceptions (part 1)

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The philosopher John Stuart Mill (1806-1873) offers an intriguing system for classifying misconceptions (or ‘fallacies’ as he terms them) that could be useful for teachers in understanding many of the misconceptions and preconceptions that our students hold.

My own thoughts on this issue have been profoundly shaped by the ‘Resources Framework‘ as presented by authors such as Andrea di Sessa, David Hammer, Edward Redish and others. What follows is not a rejection of this approach but rather an exploration of whether Mill’s work offers some relevant insights. My thought is that it quite possibly might; after all, it has happened before . . .

The authors, however, did not use or refer to Mill’s system of logic in developing the programs or in formulating their theory of instruction. They didn’t discover parallels between their theory of instruction and Mill’s logic until after they had finished writing the bulk of ‘Theory of Instruction’. The discovery occurred when they were writing a chapter on theoretical issues. In their search for literature relevant to their philosophical orientation, they came across Mill’s work and were shocked to discover that they had independently identified all the major patterns that Mill had articulated. ‘Theory of Instruction’ (1982) even had parallel principles to the methods in ‘A System of Logic’ (1843)

Engelmann and Carnine 2013: Chapter 2

Mill’s system for classifying fallacies

In A System of Logic (1843), Mill argues that

Indifference to truth can not, in and by itself, produce erroneous belief; it operates by preventing the mind from collecting the proper evidences, or from applying to them the test of a legitimate and rigid induction; by which omission it is exposed unprotected to the influence of any species of apparent evidence which offers itself spontaneously, or which is elicited by that smaller quantity of trouble which the mind may be willing to take.

Mill 1843: Book V Chap 1

Mill is saying that we don’t believe false things because we want to, but because there are mechanisms preventing our minds from duly noting and weighing the myriad evidences from which we construct our beliefs about the world by the process of induction.

He suggests that there are five major classes of fallacies:

  • A priori fallacies;
  • Fallacies of observation;
  • Fallacies of generalisation;
  • Fallacies of ratiocination; and
  • Fallacies of confusion

Erroneous arguments do not admit of such a sharply cut division as valid arguments do. An argument fully stated, with all its steps distinctly set out, in language not susceptible of misunderstanding, must, if it be erroneous, be so in some one of these five modes unequivocally; or indeed of the first four, since the fifth, on such a supposition, would vanish. But it is not in the nature of bad reasoning to express itself thus unambiguously.

Mill 1843: Book V Chap 1

Mill is saying that invalid inferences, by their very nature, are ‘messier’ and harder to classify than correct inferences. However, they must all fit into the five categories outlined above. Actually, they are more likely to fit into the first four categories since clear and unambiguous use of language and terms would tend to eliminate fallacies of confusion as a matter of course.

What is an a priori fallacy?

In philosophy, a priori means knowledge derived from theoretical deduction rather than from empirical observation or experience.

Mill says that a priori fallacies (which he also calls fallacies of simple observation) are

those in which no actual inference takes place at all; the proposition (it cannot in such cases be called a conclusion) being embraced, not as proved, but as requiring no proof; as a self-evident truth.

Mill 1843: Book V Chap 3

In other words, an a priori fallacy is an idea whose truth is accepted on its face value alone; no evidence or justification of its truth is needed. An example from physics education might be ideas such as ‘heavy objects fall’ or ‘wood floats’. Some students accept these as obvious and self-evident truths: there is no need to consider ideas such as weight and resultant force or density and upthrust because these are ‘brute facts’ about the world that admit of no further explanation. This a case of mislabelling subjective facts as objective facts.

Falling is a location-specific behaviour: objects on Earth will indeed tend to accelerate downwards towards the centre of the Earth: this is a subjective fact which is dependent on the location of the object rather than an objective fact about the behaviour of all objects everywhere (although we could, of course, argue that falling is indeed an objective fact about objects which are subject to the influence of gravitational fields). Equally, floating is not a phenomenon restricted to the interaction between wood and water: many woods will sink in low density oils. ‘Wood floats‘ is not an objective fact about the universe but rather a subjective fact about the interaction of wood with a certain liquid.

This may be why some students are incurious about certain phenomena because they regard them as trivial and obvious rather than manifestations of the inner workings of the universe.

Mill lists many other examples of the a priori fallacy, but his examples are drawn from the history of science and philosophy, and so are of less direct relevance to the science classroom, with the possible exception of the two following examples:

Humans tend to default to the assumption that any phenomenon must necessarily have only a single cause; in other words, we assume that a multiplicity of causes is impossible. We are protected from this version of the a priori fallacy by the guard rail of the scientific method. For a complete understanding of a phenomenon, we look at the effect of one independent variable at a time whilst controlling other possible variables.

There remains one a priori fallacy or natural prejudice, the most deeply-rooted, perhaps, of all which we have enumerated; one which not only reigned supreme in the ancient world, but still possesses almost undisputed dominion over many of the most cultivated minds … This is, that the conditions of a phenomenon must, or at least probably will, resemble the phenomenon itself … the natural prejudice which led people to assimilate the action of bodies upon our senses, and through them upon our minds, to the transfer of a given form from one object to another by actual moulding.

Mill 1843: Book V Chap 3

I think that this tendency might be the one in play with the difficulties that many students have with understanding how images are formed: they think that an image is an evanescent ‘clone’ of the object that is being imaged rather than being an artefact of the light rays reflected or emitted from the object. This also might help explain why students find explaining the colour changes produced by looking at an object through a colour filter or illuminating it with coloured light difficult: they assume that colour is an essential unalterable property that adheres to the object and cannot be changed without changing the object.

We’ll continue this exploration of Mill’s classification of misconceptions in later posts.

References

Engelmann, S., & Carnine, D. (2013). Could John Stuart Mill Have Saved Our Schools? Attainment Company, Inc.

Mill, J. S. (1843). A System of Logic. Collected Works.

Gravitational potential: the ‘bottom of a hole’ perspective

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The surface of Mars imaged by NASA’s Curiosity rover in 2013

April 20, 2112: The sky is flat, the land is flat, and they meet in a circle at infinity. No star shows but the big one, a little bigger than it shows through most of the [asteroid] Belt, but dimmed to red, like the sky. It’s the bottom of a hole, and I must have been crazy to risk it. […] The stars are gone, and the land around me makes no sense. Now I know why they call planet dwellers ‘flatlanders’. I feel like a gnat on a table. I’m sitting here shaking, afraid to step outside. […] I’M AT THE BOTTOM OF A LOUSY HOLE!

Larry Niven, ‘At The Bottom of a Hole’ (1966)

Redish and Kuo (2015: 586) suggest that tapping into our students’ innate physical intuitions can be a very productive teaching strategy. For example, Redish observed some physics instructors teaching non-physics majors how to interpret a potential energy U against the separation r between particles graph (diagram 8(a) below).

From Redish and Kuo (2015)

The students were finding it difficult to answer the question of whether the particles would attract or repel each other when they had energy E and were at a separation of C. Redish noted that the instructors advised the students to consider the derivative of the curve at C (diagram 8(b) above) and, since it had a positive gradient, to surmise that the force between the particles would therefore be attractive since F=-dU/dr. Redish suggested:

A more effective approach for this population might be to begin with an embodied analogy and implicitly supporting epistemologies valuing physical intuition. Start with treating a potential energy curve as a track or hill and, using the analogy of gravitational potential energy, then place a ball on the hill as shown in Fig. 8c.

Redish and Kuo (2015)

Which way would the ball roll in 8(c) roll? Redish said that the students had no problem deducing that the particles would exert an attractive force on each other at C (and a repulsive force when their energy is E at the smaller value of r) after using this analogy.

Using students’ physical intuitions to help understand gravitational potential

The episode outlined above reminded me of a science fiction story by Larry Niven that I had read many years ago. In ‘At the Bottom of a Hole’, Niven imagined what landing on a planet would feel like to a ‘Belter’; that is to say, to a human being who had spent their entire life navigating between the small worlds of the asteroid Belt: small planetoid-sized worlds whose shallow gravitational fields required only a low-intensity burn for a spaceship to slip free of their influence forever. An extract from the story is quoted as an introduction to this post: in essence, the ‘Belter’ who has lived his life voyaging between the low mass and low gravity worldlets of the asteroid belt finds it emotionally and psychologically disturbing to find himself at the bottom of a deep gravitational hole.

Gravitational Fields are always ‘holes’

Gravitational fields are always holes (unlike electric fields, of course, which can be either ‘holes’ or ‘mountains’; this may well form the basis of a later post).

The mass of the Earth produces a much deeper gravitational hole than the much smaller mass of an asteroid.

As a consequence, a spaceship near the Earth’s surface (A) needs to burn a lot more fuel (i.e. do a lot more work) to completely escape the gravitational influence of the Earth (B) then a spaceship near to the surface of an asteroid. The spaceship closer to the asteroid (C) needs a much smaller burn to completely escape its gravitational influence (D).

To a mature space-faring civilisation, living on the surface of a planet could well be likened (and seem as eccentric) as living at the bottom of a spectacularly deep hole.

Gravitational potential

The gravitational potential of an object of mass M is given by:

where G is the Gravitational Constant and r is the displacement from the centre of mass of the object. The units of V would be joules per kilogram J/kg.

Note that the magnitude of V gets larger as r decreases. This allows us to represent a gravitational field in terms of equipotential lines (dotted on the diagram below) as well as field lines (solid).

Modelling gravitational potential as a three dimensional hole

We can engage our own and our students’ physical intuitions by picturing the equipotential lines as being contour lines indicating the depth of a three dimensional hole.

An object represented as a ball at position A will not tend to roll down into the hole since there is no discernible downhill ‘slope’ at A; in effect, as r tends towards infinity then the object is beyond the effects of M’s gravity. A position outside the gravitational field of a massive object has a gravitational potential of zero.

Let’s think about what happens as r decreases until the object is at B. Here we can intuitively surmise that it will experience a small force tending to make it fall deeper into the hole. How much work will the gravitational field have done moving an object from infinity to this position? The answer is, of course, 0.5 MJ for each kilogram of mass.

How much work will be done by the gravitational field moving the object from B to C? The answer is again an extra 0.5 MJ/kg but note that this happens over a much smaller change in r than before because the gravitational field is becoming more intense. Again, we can intuit that the object will experience a stronger gravitational force at C than at B.

We can go on to argue that a similar pattern of behaviour will also occur at D and E.

But the real value of this representation is, in my opinion, helping students understand how much energy a body needs to escape the influence of a gravitational field.

If we start at B, we would have to do 0.5 MJ/kg of work on it to make it escape. In other words, it needs 0.5 MJ/kg to climb out of the hole.

If we started at C, then we would need 1.0 MJ/kg; and D, 1.5 MJ/kg and so on.

If we were considering a spacecraft operating in the vacuum of space, then transferring 2.0 MJ/kg of kinetic energy would allow ot to completely escape the gravitational influence of M; or, in other words, to reach a value of r such that its gravitational potential is zero.

Near the Earth’s surface where r = 6.38 x 106 m, the gravitational potential can be calculated as follows:

That is to say, a body would need to gain 64.4 MJ of kinetic energy for each kilogram of its mass to completely escape from the influence of the Earth’s gravity.

We can therefore calculate the escape velocity for a body near the Earth surface as follows:

As I mentioned above, I think the real power of this way of tapping into physical intuition for understanding fields comes when we use it to represent electric fields. I will cover that in a later post.

Reference

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

We all adore Caloric

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We all adore a Kia-Ora

Advertising slogan for ‘Kia-Ora’ orange drink (c. 1985)

Energy is harder to define than you would think. Nobel laureate Richard Feynman defined ‘energy’ as

a numerical quantity which does not change when something happens. It is not a description of a mechanism, or anything concrete; it is just a strange fact that we can calculate some number and when we finish watching nature go through her tricks and calculate the number again, it is the same. […] It is important to realize that in physics today, we have no knowledge of what energy is. […] It is an abstract thing in that it does not tell us the mechanism or the reasons for the various formulas.

Feynman Lectures on Physics, Vol 1, Lecture 4 Conservation of Energy (1963)

Current secondary school science teaching approaches to energy often picture energy as a ‘quasi-material substance’.

By ‘quasi-material substance’ we mean that ‘energy is like a material substance in how it behaves’ (Fairhurst 2021) and that some of its behaviours can be modelled as, say, an orange liquid (see IoP 2016).

The eight energy stores as suggested by the IoP

And yet, sometimes these well-meaning (and, in my opinion, effective) approaches can draw some dismissive comments from some physicists.

The Simpsons Comic Book Guy character saying "Picturing energy as a quasi-material substance? That teaching approach smacks of the oh-so-discredited 'Caloric' theory of energy to me . . ."
The Simpsons’ Comic Book Guy weighs in the ‘Teaching Energy’ debate

What was the ‘Caloric Theory of Energy’?

To begin with, there was never a ‘Caloric Theory of Energy’ since the concept of energy had not been developed yet; but the Caloric Theory of Heat was an important step along the way.

Caloric was an invisible, weightless and self-repelling fluid that moved from hot objects to cold objects. Antoine Lavoisier (1743-1794) supposed that the total amount of caloric in the universe was constant: in other words, caloric was thought to be a conserved quantity.

Caloric was thought to be a form of ‘subtle matter’ that obeyed physical laws and yet was so attenuated that it was difficult to detect. This seems bizarre to our modern sensibilities and yet Caloric Theory did score some notable successes.

  • Caloric explained how the volume of air changed with temperature. Cold air would absorb caloric and thus expand.
  • The Carnot cycle which describes the maximum efficiency of a heat engine (i.e. a mechanical engine powered by heat) was developed by Sadi Carnot (1796-1832) on the basis of the Caloric Theory

Why Caloric Theory was replaced

It began with Count Rumford in 1798. He published some observations on the manufacturing process of cannons. Cannon barrels had to be drilled or bored out of solid cylinders of metal and this process generated huge quantities of heat. Rumford noted that cannons that had been previously bored produced as much heat as cannons that were being freshly bored for the first time. Caloric Theory suggested that this should not be the case as the older cannons would have lost a great deal of caloric from being previously drilled.

The fact that friction could seemingly generate limitless quantities of caloric strongly suggested that it was not a conserved quantity.

We now understand from the work James Prescott Joule (1818-1889) and Rudolf Clausius (1822-1888) that Caloric Theory had only a part of the big picture: it is energy that is the conserved quantity, not caloric or heat.

As Feynman puts it:

At the time when Carnot lived, the first law of thermodynamics, the conservation of energy, was not known. Carnot’s arguments [using the Caloric Theory] were so carefully drawn, however, that they are valid even though the first law was not known in his time!

Feynman Lectures on Physics, Vol 1, Lecture 44 The Laws of Thermodynamics

In other words, the Caloric Theory is not automatically wrong in all respects — provided, that is, it is combined with the principle of conservation of energy, so that energy in general is conserved, and not just the energy associated with heat.

We now know, of course, that heat is not a form of attenuated ‘subtle matter’ but rather the detectable, cumulative result of the motion of quadrillions of microscopic particles. However, this is a complex picture for novice learners to absorb.

Caloric Theory as a bridging analogy

David Hammer (2000) argues persuasively that certain common student cognitive resources can serve as anchoring conceptions because they align well with physicists’ understanding of a particular topic. An anchoring conception helps to activate useful cognitive resources and a bridging analogy serves as a conduit to help students apply these resources in what is, initially, an unfamiliar situation.

The anchoring conception in this case is students’ understanding of the behaviour of liquids. The useful cognitive resources that are activated when this is brought into play include:

  • the idea of spontaneous flow e.g. water flows downhill;
  • the idea of measurement e.g. we can measure the volume of liquid in a container; and
  • the idea of conservation of volume e.g. if we pour water from a jug into an empty cup then the total volume remains constant.

The bridging analogy which serves as a channel for students to apply these cognitive resources in the context of understanding energy transfers is the idea of ‘energy as a quasi-material substance’ (which can be considered as an iteration of the ‘adapted’ Caloric Theory which includes the conservation of energy).

The bridging analogy helps students understand that:

  • energy can flow spontaneously e.g. from hot to cold;
  • energy can be measured and quantified e.g. we can measure how much energy has been transferred into a thermal energy store; and
  • energy does not appear or disappear: the total amount of energy in a closed system is constant.

Of course, a bridging analogy is not the last word but only the first step along the journey to a more complete understanding of the physics involved in energy transfers. However, I believe the ‘energy as a quasi-material substance’ analogy is very helpful in giving students a ‘sense of mechanism’ in their first encounters with this topic.

Teachers are, of course, free not to use this or other bridging analogies, but I hope that this post has persuaded even my more reluctant colleagues that they need a more substantive argument than a knee jerk ‘energy-as-substance = Caloric Theory = BAD’.

References

Fairhurst P. (2021), Best Evidence in Science Teaching: Teaching Energy. https://www.stem.org.uk/sites/default/files/pages/downloads/BEST_Article_Teaching%20energy.pdfhttps://www.stem.org.uk/sites/default/files/pages/downloads/BEST_Article_Teaching%20energy.pdf [Accessed April 2022]

Hammer, D. (2000). Student resources for learning introductory physics. American Journal of Physics, 68(S1), S52-S59.

Institute of Physics (2016), Physics Narrative: Shifting Energy Between Stores. Available from https://spark.iop.org/collections/shifting-energy-between-stores-physics-narrative [Accessed April 2022]

The Burnéd Hand Teaches Best

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The burned hand teaches best. After that, advice about fire goes to the heart.

J. R. R. Tolkein, The Two Towers (1954)

As is often the case in an educational context, and with all due respect to Tolkein, I think Siegfried Engelman actually said it best.

The physical environment provides continuous and usually unambiguous feedback to the learner who is trying to learn physical operations . . .

Siegfried Engelmann and Douglas Carnine, Theory of Instruction (1982)

I am going to outline a practical approach that will help students understand that black objects are good emitters and good absorbers of infrared radiation.

What I propose is a simple, inexpensive and low risk procedure (similar to this one from the IoP) that won’t actually inflict any actual “burned hands” but will, hopefully, through a clever (imho) manipulation of the physical environment, speak directly to the heart — or at least to students’ “sense of mechanism” about how the world works.

Half human and half infrared detector

Obtain tubes of matt black and white facepaint. (These are typically £5 or less.) Choose a brand that is water based for easy removal and is compliant with EU and UK regulations.

We also need a good source of infrared radiation. Some suppliers such as Nicholl and Timstar can supply a radiant heat source that is safe to use in schools. Although these can be expensive to purchase, there may already be one hiding in a cupboard in your school. If you don’t have one, use a 60W filament light bulb mounted in desk lamp (do not use a fluorescent or LED lamp — they don’t produce enough IR!). Failing that, you could use a raybox with a 24W, 12V filament lamp to act as the infrared source. [UPDATE: Paul Bushen also recommends a more economical option — an infrared heat lamp.)

Use the facepaint to make 2 cm by 2 cm squares on the back of one hand in black and in white on the other. Hold each square up to the infrared source so they are a similar distance from it.

Hold the hands still in front of the source for a set time. This could be anywhere between five seconds and a few tens of seconds, depending on the intensity of the source. You should run through this experiment ahead of time to make sure that there is minimal risk of any serious burns for the time you intend to allocate. If you are using rayboxes then you might need a separate one for each hand.

Schematic representation showing two hands with white and black paint on the back being held up to an infrared source.
The human infrared detector

The hand with the black paint becomes noticeably warmer when exposed to infrared radiation. We can deduce that this is because the colour black is better and absorbing the infrared than the white colour.

Energy is being transferred via light into the thermal energy store of the hand.

Schematic representation of energy being transferred into the thermal energy store of the hand via light.

We can use a black painted hand as a rudimentary detector for infrared. The hotter it gets, the more infrared is being emitted.

Enter Leslie’s cube . . .

Direct perception of the infrared output from a Leslie’s cube

Fill a Leslie’s cube with hot water from a kettle and then get students to place the hand with the black square a couple of centimetres away from the black face of the cube. After a few seconds, ask them to place the same hand by the white face of the cube. (Although, for the best contrast, you should maybe try the polished silver side). Make sure the student’s hand does not actually touch the face of the Leslie’s cube, otherwise they may end up with an actual burned hand!

The fact that the black face emits more infrared radiation is immediately directly perceivable by the “infrared detector” hand which feels distinctly warmer than when it’s placed next to the black coloured face rather than the white face.

This procedure is, I think, more convincing to many students as opposed to merely using (say) a digital infrared detector and reading off a larger number from the dark side compared to the white side.

Understanded of the pupils

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It is a thing plainly repugnant . . . to Minister the Sacraments in a Tongue not understanded of the People.

Gilbert, Bishop of Sarum. An exposition of the Thirty-nine articles of the Church of England (1700)

How can we help our students understand physics better? Or, in more poetic language, how can we make physics a thing that is more ‘understanded of the pupils’?

Redish and Kuo (2015: 573) suggest that the Resources Framework being developed by a number of physics education researchers can be immensely helpful.

In summary, the Resources Framework models a student’s reasoning as based on the activation of a subset of cognitive resources. These ‘thinking resources’ can be classified broadly as:

  • Embodied cognition: these are simple, irreducible cognitive resources sometimes referred to as ‘phenomenological primitives’ or p-prims such as ‘if-resistance-increases-then-the-output-decreases‘ and ‘two-opposing-effects-can-result-in-a-state-of-dynamic-balance‘. They are typically straightforward and ‘obvious’ generalisations of our lived, everyday experience as we move through the physical world. Embodied cognition is perhaps summarised as our ‘sense of mechanism’.
  • Encyclopedic (ancillary) knowledge: this is a complex cognitive resource made of a large number of highly interconnected elements: for example, the concept of ‘banana’ is linked dynamically with the concept of ‘fruit’, ‘yellow’, ‘curved’ and ‘banana-flavoured’ (Redish and Gupta 2009: 7). Encyclopedic knowledge can be thought of as the product of both informal and formal learning.
  • Contextualisation: meaning is constructed dynamically from contextual and other clues. For example, the phrase ‘the child is safe‘ cues the meaning of ‘safe‘ = ‘free from the risk of harm‘ whereas ‘the park is safe‘ cues an alternative meaning of ‘safe‘ = ‘unlikely to cause harm‘. However, a contextual clue such as the knowledge that a developer had wanted to but failed to purchase the park would make the statement ‘the park is safe‘ activate the ‘free from harm‘ meaning for ‘safe‘. Contextualisation is the process by which cognitive resources are selected and activated to engage with the issue.

Using the Resources Framework for teaching

I have previously used aspects of the Resources Framework in my teaching and have found it thought provoking and helpful to my practice. However, the ideas are novel and complex — at least to me — so I have been trying to think of a way of conveniently organising them.

What follows in my ‘first draft’ . . . comments and suggestions are welcome!

The RGB Model of the Resources Framework

The RGB Model of the Resources Framework

The red circle (the longest wavelength of visible light) represents Embodied Cognition: the foundation of all understanding. As Kuo and Redish (2015: 569) put it:

The idea is that (a) our close sensorimotor interactions with the external world strongly influence the structure and development of higher cognitive facilities, and (b) the cognitive routines involved in performing basic physical actions are involved in even in higher-order abstract reasoning.

The green circle (shorter wavelength than red, of course) represents the finer-grained and highly-interconnected Encyclopedic Knowledge cognitive structures.

At any given moment, only part of the [Encyclopedic Knowledge] network is active, depending on the present context and the history of that particular network

Redish and Kuo (2015: 571)

The blue circle (shortest wavelength) represents the subset of cognitive resources that are (or should be) activated for productive understanding of the context under consideration.

A human mind contains a vast amount of knowledge about many things but has limited ability to access that knowledge at any given time. As cognitive semanticists point out, context matters significantly in how stimuli are interpreted and this is as true in a physics class as in everyday life.

Redish and Kuo (2015: 577)

Suboptimal Understanding Zone 1

A common preconception held by students is that the summer months are warmer because the Earth is closer to the Sun during this time of year.

The combination of cognitive resources that lead students to this conclusion could be summarised as follows:

  • Encyclopedic knowledge: the Earth’s orbit is elliptical
  • Embodied cognition: The closer to a heat source you are the warmer it is.

Both of these cognitive resources, considered individually, are true. It is their inappropriate selection and combination that leads to the incorrect or ‘Suboptimal Understanding Zone 1’.

To address this, the RF(RGB) suggests a two pronged approach to refine the contextualisation process.

Firstly, we should address the incorrect selection of encyclopedic knowledge. The Earth’s orbit is elliptical but the changing Earth-Sun distance cannot explain the seasons because (1) the point of closest approach is around Jan 4th (perihelion) which is winter in the northern hemisphere; (2) seasons in the northern and southern hemispheres do not match; and (3) the Earth orbit is very nearly circular with an eccentricity e of 0.0167 where a perfect circle has e = 0.

Secondly, the closer-is-warmer p-prim is not the best embodied cognition resource to activate. Rather, we should seek to activate the spread-out-is-less-intense ‘sense of mechanism’ as far as we are able to (for example by using this suggestion from the IoP).

Suboptimal Understanding Zone 2

Another common preconception held by students is all waves have similar properties to the ‘breaking’ waves on a beach and this means that the water moves with the wave.

The structure of this preconception could be broken down into:

  • Embodied cognition: if I stand close to the water on a beach, then the waves move forward to wash over my feet.
  • Encyclopaedic knowledge: the waves observed on a beach are water waves

Considered in isolation, both of these cognitive resources are unproblematic: they accurately describes our everyday, lived experience. It is the contextualisation process that leads us to apply the resources inappropriately and places us squarely in Suboptimal Understanding Zone 2.

The RF(RGB) Model suggests that we can address this issue in two ways.

Firstly, we could seek to activate a more useful embodied cognition resource by re-contextualising. For example, we could ask students to imagine themselves floating in deep water far from the shore: do the waves carry them in any particular direction or simply move them up or down as they pass by?

Secondly, we could seek to augment their encyclopaedic knowledge: yes, the waves on a beach are water waves but they are not typical water waves. The slope of the beach slows down the bottom part of the wave so the top part moves faster and ‘topples over’ — in other words, the water waves ‘break’ leading to what appears to be a rhythmic back-and-forth flow of the waves rather than a wave train of crests and troughs arriving a constant wave speed. (This analysis is over a short period of time where the effect of any tidal effects is negligible.)

Both processes try to ‘tug’ student understanding into the central, optimal zone.

Suboptimal Understanding Zone 3

Redish and Kuo (2015: 585) recount trying to help a student understand the varying brightness of bulbs in the circuit shown.

4 bulbs in a circuit: Bulbs A, B and D are in series with the cell but bulb C is in parallel across bulb B.
All bulbs are identical. Bulbs A and D are bright; bulbs B and C are dim.

The student said that they had spent nearly an hour trying to set up and solve the Kirchoff’s Law loop equations to address this problem but had been unsuccessful in accounting for the varying brightnesses.

Redish suggested to the student that they try an analysis ‘without the equations’ and just look at the problems in simpler physical terms using just the concept of electric current. Since current is conserved it must split up to pass through bulbs B and C. Since the brightness is dependent on the current, the smaller currents in B and C compared with A and D accounts for their reduced brightness.

When he was introduced to [this] approach to using the basic principles, he lit up and was able to solve the problem quickly and easily, saying, ‘‘Why weren’t we shown this way to do it?’’ He would still need to bring his conceptual understanding into line with the mathematical reasoning needed to set up more complex problems, but the conceptual base made sense to him as a starting point in a way that the algorithmic math did not.

Analysing this issue using the RF(RGB) it is plausible to suppose that the student was trapped in Suboptimal Understanding Zone 3. They had correctly selected the Kirchoff’s Law resources from their encyclopedic knowledge base, but lacked a ‘sense of mechanism’ to correctly apply them.

What Redish did was suggest using an embodied cognition resource (the idea of a ‘material flow’) to analyse the problem more productively. As Redish notes, this wouldn’t necessarily be helpful for more advanced and complex problems, but is probably pedagogically indispensable for developing a secure understanding of Kirchoff’s Laws in the first place.

Conclusion

The RGB Model is not a necessary part of the Resources Framework and is simply my own contrivance for applying the RF in the context of physics education at the high school level. However, I do think the RF(RGB) has the potential to be useful for both physics and science teachers.

Hopefully, it will help us to make all of our subject content more ‘understanded of the pupils’.

References

Redish, E. F., & Gupta, A. (2009). Making meaning with math in physics: A semantic analysis. GIREP-EPEC & PHEC 2009, 244.

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