Siegfried Engelmann

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.

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.

Electrifying Engelmann

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It is a long-standing and melancholy truth that, despite the best efforts of many legions of Physics teachers, many students continue to not only dislike electricity, but to hate it with the white-hot intensity of a million suns.

What we have here, I think, is a classic failure to communicate.

A final fact is that samenesses and differences of examples are more obvious when the examples are juxtaposed. This fact implies that the continuous conversion of examples provides the clearest presentation of samenesses and, differences because it creates the changes that occur from one example to the next.

— Siegfried Engelmann and Douglas Carmine, Theory of Instruction (1982) p.46

Looking at my own teaching, I certainly attempt to juxtapose a number of circuits. I really want to highlight the similarities and differences between circuits in order to better develop my students’ understanding. But the problem is that both limited resources and other practical considerations mean that the juxtapositioning cannot happen by continuous conversion, except very rarely.

For example, I would set up (or ask students to set up) a circuit with a single bulb with an ammeter, then I (or we) would disassemble the circuit and rebuild it with the ammeter in a different position, or a second bulb added in series or in parallel . . .

It occurs to me that what we are relying on to thread these juxtapositions together in students’ minds is a sequence of circuit diagrams. I suppose it’s another case of the curse of knowledge writ large: experts and novices think differently.

As a beginning teacher, I remember being genuinely shocked that many students found it easier to interpret a photograph or a 3D drawing rather than the nice, clutter-free, minimalist lines of a circuit diagram.

Without a doubt, many students retain strong visual impressions of many of the circuit diagrams they encounter, but they do not parse and decode the diagrams in the same way as their teachers do.

And that, I think, is the major problem when we are introducing electric circuits.

But what to do?

— R. S. Thomas, The Cure

Can we introduce the important aspects of electrical circuits by continuous conversion of examples?

I think we can. And what is more, I think it will be more effective than the itty-bitty assembly and disassembly of circuits that I have practiced to date.

Conservation of electrical current (and current in parallel circuits) by continuous conversion

Parallel Circuit

This is introduced with a teacher demonstration of the above circuit. Students are invited to note the identical readings on both ammeters and asked to explain why they are identical. They are then asked to predict the effect of adding a second bulb in parallel. The teacher then adds the second bulb by connecting the flying lead. The process is repeated with the third and fourth bulbs, with the teacher testing students’ understanding by asking them to predict the change in current readings as bulbs are added and removed. The teacher also tests students’ understanding of the conservation of current by asking students to predict whether the reading on both ammeters will be the same or different as bulbs are added and removed.

I find it useful to include a bulb that is not identical to the other three. It should be noticeably brighter or dimmer than the other three with the same p.d. so that students do not make the incorrect inference that the current always increases or decreases in equal steps when the circuit is changed.

The teacher could also draw the original circuit on a student whiteboard and ask students to do likewise. The changes that are about to be made could be described and students could be asked could alter the picture/circuit diagram and write their prediction on their whiteboards. They could then compare their version with the teacher’s and their prediction could be quickly tested by making the proposed changes “live” in front of the students.

If resources and time permit, students could then, of course, go on to construct their own parallel circuits as a class practical. However, I think it is important that these vital, foundational ideas are introduced (or re-introduced!) via a teacher demonstration to avoid possible cognitive overload for students.

Series circuits by continuous conversion

Series Circuit

In this demonstration circuit, four of the three bulbs are short-circuited so that they are initially unlit. The teacher asks students to explain only one bulb in the circuit is lit: it is helpful if they have previously encountered parallel circuits and can explain this in terms of electrical current taking the “easier” route (assuming they have not yet encountered the concept of electrical resistance).

Again, the two ammeters allow the teacher to emphasise and test students understanding of the idea that current is conserved.

The teacher then asks students to predict the change in current reading when switch X is opened: will it increase or decrease? Why would it increase or decrease? The process is repeated with switches Y and Z and students’ understanding is tested by asking them to predict the effect on the current reading of opening or closing X, Y or Z.

As before, the teacher would amend her circuit diagram on her student whiteboard and students would do likewise. For example: “I am going to open switch Y. Change the circuit diagram. Show me. What will happen to the reading on the left hand ammeter? What will happen to the reading on the right hand ammeter? Explain why.”

Again, I recommend that at least one out of the four bulbs in not identical to the other three to help prevent students from drawing the incorrect inference that the current will always increase or decrease in identical steps.

Engelmann (and John Stuart Mill) Revisited

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Even for the most enthusiastic and committed of us, Engelmann and Carnine’s Theory of Instruction (1982) is a fabulously intimidating read.

I have written about some of the ideas before, but a recent conversation with a fellow Physics teacher (I’m looking at you, @DeepGhataura) suggested to me that a revisit might be in order.

In a nutshell, we were talking about sets of examples. Engelmann and Carnine argue that learners learn when they construct generalisations or inferences from sets of examples. It is therefore essential that the sets of examples are carefully chosen and sequenced so that learners do not accidentally generate false inferences. A “false inference” in this context is any one that the instructor does not intend to communicate.

Engelmann and Carnine painstakingly constructed a set of logical rules that they hoped would minimise (or, more ambitiously, completely eliminate) the possibility of generating false inferences. These include the sameness principle of juxtaposition and the difference principle of juxtaposition.

However, in 2011 Carnine and Engelmann realised that they had, in a sense, been re-inventing the wheel as the same logical rules had been formulated by philosopher John Stuart Mill in A System of Logic (1843).

They outlined their system using Mill’s terms and language in the book Could John Stuart Mill Have Saved Our Schools? (2011).

 

The Method of Difference (The Difference Principle of Juxtaposition)

How can we use examples to communicate a concept to learners so that the possibility of their drawing false inferences is minimised?

The Method of Difference seeks to establish the limits of a given concept A by explicitly considering not-A.

Imagine a learner who did not understand the concept of blue. We would introduce the concept by showing (say) a picture of a blue bird and saying “This is blue.” We would then show a picture of a bird identical in every respect except that it’s colour was (say) green and say “This is not blue.”

So-far-so-blindingly-obvious, you might say. What you might not immediately appreciate is that applying this simple method rules out a large set of possible misconceptions. Without explicitly considering not-A, a learner might, with some justification, conclude that blue meant “has a beak” or “has feathers”. The Method of Difference rules out these false inferences.

Mind your P’s and Q’s

For a beginning reader, the letters p, q, b and d are problematic since they all share the same basic shape. The difference between them is a difference of orientation. Carnine and Engelmann suggest writing the letter ‘p’ on a transparent sheet and rotating and flipping the sheet to explicitly teach the difference between p and not-p.

PND

PNB

PNQ

Could this be used in Physics teaching?

 

Don’t zig when you ought to zag

Possibly — one recurring problem that I’ve noticed is that some A-level students routinely mix up magnetic and electric fields. They apply Coulomb’s Law when they should be applying F = BIl , and apply Fleming’s Left Hand Rule where it has no business being applied.

It seems reasonable to assume that it is not a lack of knowledge that is holding them back, but rather a misapplication of knowledge that they already possess. In other words, they are drawing the wrong inference from the example sets that have been presented to them.

Could using the Method of Difference at the beginning of the teaching sequence stop learners from drawing false inferences about the nature of electric and magnetic fields?

BandEfields1

BandEfields2

You know, I rather think it might…

Practicals Make Perfect

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The physical environment provides continuous and usually unambiguous feedback to the learner who is trying to learn physical operations, but does not respond to the learning attempts for cognitive operations.

Engelmann, Siegfried; Carnine, Douglas. Theory of Instruction: Principles and Applications (Kindle Locations 1319-1320). NIFDI Press. Kindle Edition.

Into The Dustbin Of Pedagogy?

Helen Rogerson asks: “Should we bin [science] practicals?” and then answers emphatically: “No. We should get better at them.”

I wholeheartedly concur with her last statement, but must confess that I find it hard to articulate why I feel practical science is such a vital component of science education.

The research base in favour of practical science is not as clear cut as one would wish, as Helen points out in her blog.

New-kid-on-the-blog Adam Boxer has even written a series of blog posts with the provocative title “Teaching Practical Skills: Are We Wasting Our Time?“. He writes:

[T]his then raises the question of “what about the kids who are never going to see a pipette dropper again once they’ve left school?” I don’t have a great answer to that. Even though all knowledge is valuable, it comes with an opportunity cost. The time I spend inculcating knowledge of pipette droppers is time I am not spending consolidating knowledge of the conservation of matter or evolution or any other “Big Idea.” [ . . . ] But if you’ve thought about those things, and you and your department conclude that we do need to teach students how to use a balance or clamp stand or Bunsen burner, then there is no other way to do it – bring out the practical! Not because anyone told you to, but because it is the right thing for your students.

Broadly positive, yes. But am I alone in wishing for a firmer foundation on which to base the plaintive mewling of every single science department in the country, as they argue for a major (or growing) share of ever-shrinking resources?

The Wrong Rabbit Hole

Image result for rabbit hole

I think a more substantive case can indeed be made, but it may depend on the recognition that we, as a community of science teachers and education professionals, have gone down the wrong rabbit hole.

By that, I think that we have all drunk too deep of the “formal investigation” well, especially at KS3 and earlier. All too often, the hands-on practical aspect plays second- or even third- or fourth-fiddle to the abstract formalism of manipulating variables and the vacuous “evaluation” of data sets too small for sound statistical processing.

So, Which Is The Right Rabbit Hole?

The key to doing science practicals “better” is, I think, to see them as opportunities for students to get clear and unambiguous feedback about cognitive operations from the physical environment.

To build adequate communications, we design operations or routines that do what the physical operations do. The test of a routine’s adequacy is this: Can any observed outcome be totally explained in terms of the overt behaviours the learner produces? If the answer is “Yes,” the cognitive routine is designed so that adequate feedback is possible. To design the routine in this way, however, we must convert thinking into doing.

Engelmann, Siegfried; Carnine, Douglas. Theory of Instruction: Principles and Applications (Kindle Locations 1349-1352). NIFDI Press. Kindle Edition.

Angle of Incidence = Angle of Reflection: Take One

It’s a deceptively simple piece of science knowledge, isn’t it? Surely it’s more or less self-evident to most people…

How would you teach this? Many teachers (including me) would default to the following sequence as if on autopilot:

  1. Challenge students to identify the angle of incidence as the independent variable and the angle of reflection as the dependent variable.
  2. Explain what the “normal line” is and how all angles must be measured with reference to it.
  3. Get out the rayboxes and protractors. Students carry out the practical and record their results in a table.
  4. Students draw a graph of their results.
  5. All agree that the straight line graph produced provides definitive evidence that the angle of reflection always equals the angle of incidence, within the limits of experimental error.

Image result for angle of incidence reflection graph
Image Credit

I’m sure that practising science teachers will agree that Stage 5 is hopelessly optimistic at both KS3 and KS4 (and even at KS5, I’m sorry to say!). There will be groups who (a) cannot read a protractor; (b) have used the normal line as a reference for measuring one angle but the surface of the mirror as a reference for the other; and (c) every possible variation of the above.

The point, however, is that this procedure has not allowed clear and unambiguous feedback on a cognitive operation ( i = r) from the physical environment. In fact, in our attempt to be rigorous using the “formal-investigation-paradigm” we have diluted the feedback from the physical environment. I think that some of our current practice dilutes real-world feedback down to homeopathic levels.

Sadly, I believe that some students will be more rather than less confused after carrying out this practical.

Angle of Incidence = Angle of Reflection: Take Two

How might Engelmann handle this?

He suggests placing a small mirror on the wall and drawing a chalk circle on the ground as shown:

engelmannreflection
Theory of Instruction (Kindle Location 8686)

Initially, the mirror is covered. The challenge is to figure out where to stand in order to see the reflection of an object.

engelmannreflection2

Note that the verification comes after the learner has carried out the steps. This point is important. The verification is a contingency, so that the verification functions in the same way that a successful outcome functions when the learner is engaged in a physical operation, such as throwing a ball at a target. Unless the routine places emphasis on the steps that lead to the verification, the routine will be weak. [ . . . ]

If the routine is designed so the learner must take certain steps and figure out the answer before receiving verification of the answer, the routine works like a physical operation. The outcome depends on the successful performance of certain steps.

Engelmann, Siegfried; Carnine, Douglas. Theory of Instruction: Principles and Applications (Kindle Locations 8699-8709). NIFDI Press. Kindle Edition.

Do I want to abandon all science investigations? Of course not: they have their place, especially for older students at GCSE and A-level.

But I would suggest that designing practical activities in such a way that more of them use the physical environment to provide clear and unambiguous feedback on cognitive ideas is a useful maxim for science teachers.

Of course, it is easier to say than to do. But it is something I intend to work on. I hope that some of my science teaching colleagues might be persuaded to do likewise.

A ten-million year program in which your planet Earth and its people formed the matrix of an organic computer. I gather that the mice did arrange for you humans to conduct some primitively staged experiments on them just to check how much you’d really learned, to give you the odd prod in the right direction, you know the sort of thing: suddenly running down the maze the wrong way; eating the wrong bit of cheese; or suddenly dropping dead of myxomatosis.

Douglas Adams, The Hitch-Hiker’s Guide To The Galaxy, Fit the Fourth

Engelmann and Direct Instruction (Part 6)

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In Could John Stuart Mill Have Saved Our Schools?, Siegfried Engelmann and Douglas Carnine discuss the philosophical foundations of their acclaimed Direct Instruction programme (see Part 1). They write of their serendipitous rediscovery of Mill’s work and that 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 [1991] even had parallel principles to the methods in [Mill’s] A System of Logic [published in 1843].

— location 543 Kindle edition

I think it’s worth looking in detail at the five principles of inference proposed by Mill, and how Engelmann and Carnine adapted them for use in educational contexts.

The five principles put forward by Mill are:

  1. The Direct Method of Agreement
  2. The Method of Difference
  3. The Joint Method of Agreement and Difference
  4. The Method of Residues
  5. The Method of Concomitant Variations

In this post, I will focus on the first three.

methodofagreement

“Non-canonical” statements and inferences are highlighted with an asterisk. These are my own suppositions and, although I believe they are supported by my reading of Mill’s and Engelmann’s and Carnine’s work, I cannot claim that they have direct textual support.

To further explain the simplified “symbolic form” I have developed to highlight what I think are the salient features in the argument:

  • A= “blue”
  • a = is blue
  • b = has beak
  • c = has wings
  • d = extends above the horizon
  • e = has clouds
  • f  = has windscreen
  • g = has four wheels

Please note that the “symbolic form” is currently only a shorthand system, and any resemblance to the notation of formal symbolic logic is merely coincidental.

methodofdifference

Links to Part 5 and the other parts of the series can be found here.

Part 1    Part 2     Part 3     Part 4    Part 5

Engelmann and Direct Instruction (Part 5)

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‘Sunlight’s a thing that needs a window
Before it enter a dark room.
Windows don’t happen.’

— R. S. Thomas, “Poetry For Supper”

For this post, I have decided to dispense with the abstract logic-chopping of some of the earlier posts in this series. (Although, I confess, I am quite partial to a nice bit of abstruse ratiocination now and again — in moderation, of course.)

Instead, I want to focus on what a teaching sequence using the principles outlined by Engelmann would actually look like in practice.

 
Some Basics

Firstly, the designer must have an expert-level understanding of the content to be taught.

[If] we are to understand how to communicate a particular bit of knowledge (such as knowledge of the color red, or knowledge about the operation of square root), we must understand the essential features of the particular concept that we are attempting to convey. [1]

The analysis of the knowledge system assumes that the designer will be able to create efficiency in what is to be taught if the designer understands the technically relevant details of the content that is to be taught. [2]

Secondly, a Direct Instruction sequence should be efficient; that is to say, it will aim to produce significant.results with the minimum effort.

The efficiency results from teaching only the skills and strategies that are necessary, and from designing strategies that apply to large segments of what is to be taught, rather than small segments. [2]
The goal is simply to teach as little as possible to provide thorough coverage of the content.[3]

Thirdly, there is no single “royal road” for Direct Instruction: two designers may map out entirely different routes while still being consistent with the guiding principles of D.I.:

[T]his efficiency does not imply general strategies for teaching something like beginning reading, critical analysis, or pre-algebra. As also noted, there is no single right way to achieve this efficiency; however, there are ways that are more efficient than others. [3]

 

Order and Efficiency

The guiding principles for ensuring efficiency are as follows:

[A]rrange the order of introduction of things to be taught for a particular topic or operation so that the more generalizable parts are taught first, and the exceptions or details that have limited application are introduced later. [3]

[However each] exception must be taught because if it is ignored, the learner may not learn it. [4]

The most efficient arrangement is to teach something and then [practice and review it] at a high rate.. . . Once taught, the operation should be used regularly. [5]

When a teaching sequence is developed using these principles, it may look very different from more familiar teaching sequences. For example, in a sequence for teaching basic fractions developed by Engelmann, the terms “top number” and “bottom number” rather than “numerator” and “denominator” were used.

The rationale for not using the “technical terms” is that they do not facilitate the instruction in any way, and they logically complicate the teaching by introducing a discrimination that is irrelevant to understanding fractions. [6]

Carnine and Engelmann argue that this allows learners to focus on what the numbers do rather than on what they are called. They are very insistent, however, that the correct technical terms will be taught — but not necessarily at the beginning as in a standard course.

This has led to some teaching sequences that are significantly different from the ones that most teachers are familiar with. Carnine and Engelmann comment that:

The point is that something may look quite simple but requires significant care in teaching, while something else (like the fraction relationship) may appear to be quite abstract but is quite easy to teach. The difficulty of what is taught is judged by the performance of students who are learning the material. [7]

Since D.I. courses seek to group together irregularities that are irregular in similar ways, Carnine and Engelmann say that this

. . . results in efficiency, but it may create a set of examples that are traditionally not grouped together. [8]

 

Thinking Into Doing

A maths teacher friend challenges a student who is intimidated by some difficult new learning with the question “Tell me what’s the most difficult thing that you’ve ever learned how to do?”

In my friend’s opinion, the most difficult thing that most people have learned is how to walk. He then goes on to assure the worried student that the same techniques that allowed her to learn to gad about on two feet from an early age will serve her well in maths (e.g. not giving up after the first fall, not minding looking a bit silly at times, learning from your mistakes, and so on).

I think it’s a nice analogy that can help students, and I’ve shamelessly lifted this tactic from him. However, as Carnine and Engelmann point out, learning a physical operation such as walking has one major advantage over learning all cognitive operations. The advantage is that the physical environment usually provides immediate, continuous and unambiguous feedback on physical operations.

The physical environment, when viewed as an active agent, either prevents the learner from continuing or provides some unpleasant consequences for the inappropriate action.. . . [However the] physical environment does not provide feedback when the learner is engaged in cognitive operations. . . . To build adequate communications, we design operations or routines that do what the physical operations do. The test of a routine’s adequacy is this: Can any observed outcome be totally explained in terms of the overt behaviours the learner produces? If the answer is “Yes,” the cognitive routine is designed so that adequate feedback is possible. To design the routine in this way, however, we must convert thinking into doing. [9]

Thinking Into Doing
An example of Thinking Into Doing? From Theory of Instruction (1982) location 1273

The aim of Direct Instruction is to provide a measure of immediate, continuous and immediate feedback for cognitive operations that is analogous to that provided by the physical environment for physical operations.
Another One In The Eye For Traditional Differentiation?

Direct Instruction stimulus material is meant to be carefully designed so as to be logically unambiguous. It should generate one — and only one — inference for all learners. This means that as long as students respond correctly to the material, we can assume that both high and low performers have learned the same inference:

The major difference between higher and lower performers is the rate at which they learn the material, not the way they formulate inferences. This difference does not support designing one sequence for higher performers and another for lower performers, but rather providing more repetition and practice for the lower performers. [10]

I don’t know about you, but to me this sounds absolutely great. If I may borrow a phrase from my fellow blogger, The Quirky Teacher: who’s with me?

 

Links

To access the previous blogposts in this series, click on the links:

Part 1    Part 2     Part 3     Part 4

Part 6 can be found here.

 

References

[1] Carnine, D. and Engelmann, S., Theory of Instruction: Principles and Applications (1982), Kindle location 299

[2] Carnine, D. and Engelmann, S., Could John Stuart Mill Have Saved Our Schools? (2011), Kindle location 610

[3] 2011 loc 640

[4] 2011 loc 678

[5] 2011 loc 671

[6] 2011 loc 648

[7] 2011 loc 671

[8] 2011 loc 685

[9] 1982 loc 1319-1348

[10] 2011 loc 719

Engelmann and Direct Instruction (Part 4)

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Much is due to those who first broke the way to knowledge, and left only to their successors the task of smoothing it.

— Samuel Johnson, A Journey To The Western Isles Of Scotland (1775)

In 1982 Siegfried Engelmann and Douglas Carnine published their Theory of Instruction. This was some 300 years after Newton started the scientific revolution by publishing his Principia; and some 70 years after Russell and Whitehead in the Principia Mathematica attempted to show that the entirety of mathematics could be derived from the laws of logic (famously taking 300 pages to prove that 1+1=2).

In short, Engelmann and Carnine were attempting to start an educational scientific revolution. They wanted to replace the traditional liberal arts foundation of educational theory with a rigorously logical scientific foundation. Their Theory of Instruction is quite simply nothing less than an attempt to write a Principia Pedagogica.

Towards a 'Principia Pedagogica'?

Effective instruction is not born of grand ideas or scenarios that appeal to development or love of learning. It is constructed from the logic and tactics of science.

— S. Engelmann and D. Carnine, Could John Stuart Mill Have Saved Our Schools? Kindle edition, location 1944

In the opening section of the T.O.I., Carnine and Engelmann argue that human beings learn primarily, and in fact literally, from the power of example.

[Learners have the] capacity to learn any quality that is exemplified through examples (from the quality of redness to the quality of inconsistency) . . . This mechanism . . . is capable of learning qualities as subtle as the unique tone of a particular violin or qualities that involve the correlation of events (such as the relationship of events on the sun to weather on the earth).

— S. Engelmann and D. Carnine, Theory of Instruction: Principles and Applications, Kindle edition, locations 365-383

To this end, they propose a simplified (or minimalist, if you will) two step mechanism of how human beings learn:

The first step of the proposed learning mechanism is the presentation of a range of examples. For instance, to explain the concept of “red” an instructor would present examples of red objects; to explain the concept of “conservation of volume”, she would present instances of (say) a fixed volume of liquid being poured into containers of varying shapes.

The second step of the learning mechanism is when the learner mentally constructs a valid generalisation, or mental rule about the qualities or features common to the examples presented. Carnine and Engelmann argue that human beings naturally and immediately attempt to generalise or form such rules when presented with any new information.

Table
From: T.O.I, Kindle loc 399

They do not attempt to argue that the whole of human behaviour can be reduced to this simple two step mechanism, but merely that by accepting this simple model, one “can account for nearly all observed cognitive behaviour” (T.O.I. Kindle loc 375)

Note that the first step is about what is to be learned, and the second step is about how it is learned.

In Carnine and Engelmann’s view, the first step is the responsibility of the instructor. The planning should focus on a rigorous logical analysis of the concept that is to be taught, and should not include any consideration of the likely behavioural response of the learner (e.g. whether she will find it “fun”).

The only factor that limits the learner . . . is the acuity of the sensory mechanism that receives information about [the concept]. (T.O.I. Kindle loc 380-1)

The second step is within the purview of the learner. However, this is also the point at which the instructor would use behavioural analysis to ascertain

. . . the extent to which the learner does or does not possess the mechanisms necessary to respond to the . . . presentation of the concept. [Then one should design] instruction for the unsuccessful learner that will modify the learner’s capacity to respond to the . . . presentation. This instruction is not based on a logical analysis of the communication, but on a behaviour analysis of the learner. (T.O.I. Kindle loc 348-352)

Teachers will, of course, be aware that this is not how we do things in our current educational system, especially as far as the standard techniques of differentiation are concerned.

Are Carnine and Engelmann correct? I’m not sure, but I find their ideas fascinating. Looking at them through the lens of my experience, I would go so far as to say that, intuitively at least, they appear to have the copper-bottomed “ring of truth” (as Richard Ingrams used to say) about them. At the very least, they are deserving of further study and attention.

 Links

To access the other blogposts in this series, click on the links:

Part 1    Part 2     Part 3    Part 5

 

To Script Or Not To Script

e=mc2andallthat2 Comments

Many moons ago — when I was younger, fitter and slimmer — I was lucky enough to attend a martial arts seminar with a famous martial arts master. His attitude and practice was anything but the mystical tosh spouted by some of his movie equivalents. Rather, his focus was intensely pragmatic: he had studied martial arts all over the world in order to find out what worked. This focus had been started by his early U.S. “police judo” training: if someone comes at you like this then you try this or this or this.

The reason I mention this that many teachers seem to hate the idea of a scripted lesson. To my mind, a scripted lesson doesn’t necessarily entail a series of reductive, robotic responses. Instead, it could simply be a list of suggested sequences and interactions that have been found to work historically.

I often — perhaps too often — write and use my own resources (“Use another teacher’s resources? Ugh! I’d rather use their toothbrush…! “)
But even I, the king of the animated PowerPoint and the Amadeus of the well-crafted worksheet, would consider using an externally written script if it demonstrably worked.

Whether Direct Instruction scripts really, really work in this sense, I can’t say. The proof of the pudding will be in the eating, or in the teaching.

And, as any actor could tell you, even the simplest script can still leave the actor with a wealth of choices…

Engelmann and Direct Instruction (Part 3)

e=mc2andallthat15 Comments

I’m going to begin this post by pondering a deep philosophical conundrum (hopefully, you will find some method in my rambling madness as you read on): I want to discuss the meaning of meaning.

image
Image from https://www.flickr.com/photos/christiaan_tonnis/15768628869

Ludwig Wittgenstein begins the Philosophical Investigations (1953), perhaps one of the greatest works of 20th Century philosophy, by quoting Saint Augustine:

When they (my elders) named some object, and accordingly moved towards something, I saw this and I grasped that the thing was called by the sound they uttered when they meant to point it out. Their intention was shewn by their bodily movements . . . I gradually learnt to understand what objects they signified; and after I had trained my mouth to form these signs, I used them to express my own desires.
— Confessions (397 CE), I.8

image
Image from https://commons.m.wikimedia.org/wiki/File:Antonio_Rodríguez_-_Saint_Augustine_-_Google_Art_Project.jpg

Wittgenstein uses it to illustrate a simple model of language where words are defined ostensively i.e. by pointing. The method is, arguably, highly effective when we wish to define nouns or proper names. However, Wittgenstein contends, there are problems even here.

If I hold up (say) a pencil and point to it and say pencil out loud, what inference would an observer draw from my action and utterance?

image

They might well infer that the object I was holding up was called a pencil. But is this the only inference that a reasonable observer could legitimately draw?

The answer is a most definite no! The word pencil could, as far as the observer could tell from this single instance, mean any one of the following: object made of wood; writing implement; stick sharpened at one end; piece of wood with a central core made of another material; piece of wood painted silver; object that uses graphite to make marks, thin cylindrical object, object with a circular or hexagonal cross-section . . . and many more.

The important point is that one is not enough. It will take many repeated instances of pointing at a range of different pencil-objects (and perhaps not-pencil-objects too) before we and the observer can be reasonably secure that she has correctly inferred the correct definition of pencil.

If defining even a simple noun is fraught with philosophical difficulties, what hope is there for communicating more complicated concepts?

Siegfried Engelmann suggests that philosopher John Stuart Mill provided a blueprint for instruction when he framed formal rules of inductive inference in A System of Logic (1843). Mill developed these rules to aid scientific investigation, but Engelmann argues strongly for their utility in the field of education and instruction. In particular, they show “how examples could be selected and arranged to form an example set that generates only one inference, the one the teacher intends to teach.” [Could John Stuart Mill Have Saved Our Schools? (2011) Kindle edition, location 216, emphasis added].

Engelmann identifies five principles from Mill that he believes are invaluable to the educator. These, he suggests, will tell the educator:

how to arrange examples so that they rule out inappropriate inferences, how to show the acceptable range of variation in examples, and how to induce understanding of patterns and the possible effects of one pattern on another. [loc 223, emphasis added]

Engelmann considers Mill’s Method of Agreement first. (We will look at the other four principles in later posts.)

Mill states his Method of Agreement as follows:

If two or more instances of the phenomenon under investigation have only one circumstance in common, the circumstance in which alone all the instances agree, is the cause (or effect) of the given phenomenon.
— A System of Logic. p.263

Engelmann suggests that with a slight change in language, this can serve as a guiding technical principle that will allow the teacher to compile a set of examples that will unambiguously communicate the required concept to the learner, while minimising the risk that the learner will — Engelmann’s bête noire! — draw an incorrect inference from the example set.

Stated in more causal terms, the teacher will identify some things with the same label or submit them to the same operation. If the examples in the teaching set share only one feature, that single feature can be the only cause of why the teacher treats instances in the same way. [Loc 233]

As an example of an incorrect application of this principle, Engelmann gives the following example set commonly presented when introducing fractions: 1/2, 1/3, and 1/4.

Engelmann argues that while they are all indeed fractions, they share more than one feature and hence violate the Method of Agreement. The incorrect inferences that a student could draw from this set would be: 1) all fractions represent numbers smaller than one; 2) numerators and denominators are always single digits; and 3) all fractions have a numerator of 1.

A better example set (argues Engelmann) would be: 5/3, 1/4, 2/50, 3/5, 10/2, 1/5, 48/2 and 7/2 — although he notes that there are thousands more possible sets that are consistent with the Method of Agreement.

Engelmann comments:

Yet many educators believe that the set limited to 1/2, 1/3, and 1/4 is well conceived. Some states ranging from North Dakota to Virginia even mandate that these fractions should be taught first, even though the set is capable of inducing serious confusion. Possibly the most serious problem that students have in learning higher math is that they don’t understand that some fractions equal one or are more than one. This problem could have been avoided with early instruction that introduced a broad range of fractions. [Loc 261]

For my part, I find Engelmann’s ideas fascinating. He seems to be building a coherent philosophy of education from what I consider to be properly basic, foundational principles, rather than some of the “castles in the air” that I have encountered elsewhere.

I will continue my exploration of Engelmann’s ideas in subsequent posts. You can find Parts 1 and 2 of this series here and here.

The series continues with Part 4 here.