## Electrifying Engelmann

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

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

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

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.

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.

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?

You know, I rather think it might…

## Practicals Make Perfect

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

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.

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:

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

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)

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.

“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.

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

## Engelmann and Direct Instruction (Part 5)

‘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]

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?

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

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)

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.

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.

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.

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

## To Script Or Not To Script

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)

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.

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

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?

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.

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.

## Engelmann and Direct Instruction (Part 2)

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

What Engelmann and Carnine are attempting to do is no less than develop a scientifically reliable model of education. In their view, learners learn by constructing inferences based on the evidence or examples presented by the teacher. In other words, learners use the rules of reason and logic (consciously or unconsciously) to develop general principles from specific examples by inductive reasoning.

To me, this is a fascinating idea. Have Engelmann and Carnine hit upon the elusive essence of what learning is? Is learning genuinely a matter of constructing inferences from evidence by formal or informal logical rules?

My view is that it certainly seems a plausible idea. In the light of my own experience and thinking it has a “ring of truth”, and I suspect that I am going to find this a profoundly influential idea for the rest of my career.

Many authors and thinkers have argued that human beings construct “mental maps” or conceptual models constructed by inductive reasoning from often limited information. Anthropologist Louis Liebenberg describes an example involving the !Xõ people of the central Kalahari Desert:

While tracking down a solitary wildebeest spoor [tracks] of the previous evening !Xõ trackers pointed out evidence of trampling which indicated that the animal had slept at that spot. They explained consequently that the spoor leaving the sleeping place had been made early that morning and was therefore relatively fresh. The spoor then followed a straight course, indicating that the animal was on its way to a specific destination. After a while, one tracker started to investigate several sets of footprints in a particular area. He pointed out that these footprints all belonged to the same animal, but were made during the previous days. He explained that the particular area was the feeding ground of that particular wildebeest. Since it was, by that time, about mid-day, it could be expected that the wildebeest may be resting in the shade in the near vicinity.

— quoted by Steven Pinker in How The Mind Works p. 193

The trackers were using miniscule traces of evidence and their knowledge of the environment to make inferences about the behaviour of (currently) unseen entities. In other words, they were using inductive reasoning to put together a tentative model of what their quarry was doing or attempting to do. (And I use ‘tentative’ in the sense that the model will be adapted and corrected in the light of further evidence.)

As do we all! I would suggest that all humans use similar techniques of inference, or ‘mental modules’ in Steven Pinker’s memorable phrasing, even with vastly different subject matter. Stephen Hawking and Leonard Mlodinow even go so far as to suggest that:

we shall adopt an approach that we call model-dependent realism. It is based on the idea that our brains interpret the input from our sensory organs by making a model of the world. When such a model is successful at explaining events, we tend to attribute to it, and to the elements and concepts that constitute it, the quality of reality.

The Grand Design p.9

And where does this leave us? If Engelmann and Carnine are correct (and I believe they are} then education becomes a matter of logic. They argue that a vital criterion in designing what they call “sound instructional sequences” is that sets of examples should “generate only the intended inferences”. They note

that logical flaws in instruction could be identified analytically, through a careful examination of the teaching. If we know the specific set of examples and the inference that the learners are supposed to derive from the instruction, we can determine if serious false inferences are implied by the program.

— location 1514

And I, for one, find that a highly engaging and strangely comforting thought.

(You can read Part 3 here)

## Engelmann and Direct Instruction (Part 1)

We are art’s mercenaries,
firing our thought’s arrows
at the mystery of things
— R. S. Thomas, Paving

Engelmann comes highly recommended:

In his book Visible Learning: A Synthesis of Over 800 Meta-Analyses Relating to Achievement, the researcher John Hattie evaluates the success of a range of different teaching approaches. As the subtitle suggests, he synthesised the results of hundreds of different analyses of achievement and measured the effect of different factors . . . A specific Direct Instruction programme was developed by the American educator, Siegfried Engelmann, in the 1960s. It proved incredibly successful but also incredibly controversial because it contradicted so much of what theorists like Dewey and Freire advocated. Hattie specifically endorsed Engelmann’s programme.

— Daisy Christodoulo, Seven Myths About Education, location 751 Kindle edition

Later on in the book, Hattie confronts the dominance of empirically unsuccessful constructivist ideas in teacher training. He explains the effectiveness of Direct Instruction, a structured and unapologetically teacher-led method of instruction originated in 1960s America. Despite being shunned by the American education establishment, Hattie’s analysis shows that Direct Instruction has one of the largest effect sizes (0.59) for any teaching programme.

— Robert Peal, Progressively Worse, location 2689 Kindle edition

I was intrigued and wanted to find out more, so I recently read Siegfried Engelmann’s and Douglas Carnine’s book Could John Stuart Mill Have Saved Our Schools? which can be thought of as an introduction to the philosophical underpinning of Direct Instruction.

I claim no particular expertise in this field beyond that of a working teacher with a couple of decades of experience. I suppose that it is also appropriate at this point to disclose that my practice generally tends towards the traditional-didactic rather than the progressive end of the spectrum, so I am perhaps predisposed to be sympathetic to Engelmann’s ideas. Nevertheless, this blog will attempt to engage critically with his ideas and arguments.

Engelmann and Carnine open by saying that (unfortunately, in their opinion) education has historically been excluded from the domain of science. They suggest that the five principles of induction put forward by philosopher John Stuart Mill in his A System Of Logic (1843) would form a suitable basis for a scientific systematisation of effective educational practice. The efficacy of these principles when applied to education was not recognised at the time, not even by Mill himself, until Engelmann and Carnine rediscovered them in the 1970s.

I was unfamiliar with this aspect of Mill’s work, and it was a delight to be introduced to it. I was particularly struck by this bombshell from Mill:

In another of its senses, to reason is simply to infer any assertion, from assertions already admitted: and in this sense induction is as much entitled to be called reasoning as the demonstrations of geometry
— J. S. Mill, A System of Logic, location 175 Kindle edition

Philosophers have long debated the “problem of induction”. It is generally recognised that deductive reasoning (e.g. Socrates is a man; all men are mortal; therefore Socrates is mortal) is more dependable that inductive reasoning (e.g. every swan I have seen to date has been white; therefore every swan I will see in the future will be white).

However, it is a under-acknowledged truth that in our day-to-day lives (and in science generally) we rely primarily on induction and inference and, for the most part, they serve us well. What Mill is attempting to do is address the philosophical “second class status” accorded to inductive truths by formalising a set of rules that allow us to generate valid inductive inferences.

Engelmann and Carnine argue that these rules are of fundamental importance to the teacher as they allow her to construct a system of instruction that allows students to generate valid inferences and minimise false inferences:

In summary, the fabric of well designed instruction consists of details that promote specific inferences and rule out inappropriate inferences. 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? location 1944 Kindle edition

One example they present is that of a constructivist approach to the teaching of prime numbers by getting students to lay out numbers of beans in rows and columns: students are invited to notice that some numbers (e.g. 7) cannot be laid out in rows of more than one bean which have equal numbers of beans. Englemann and Carnine argue that this activity does not accord with Mill’s principles because it will encourage students to generate a number of false inferences:

The false inference is that prime numbers are odd numbers. Imagine the consternation of the student who later discovers that 9 and 15 are odd, but they generate multiple rows. In contrast, 2 is even, but it is prime. A related false inference is that there is some form of predictable pattern for the occurrence of prime numbers, rather than the fact that some numbers are primes and others aren’t. Unless students had received previous instruction on what primes are, the bean counting has a potential of inducing false inferences; however, if students first learn the properties of prime numbers, the bean counting is a pointless activity. It simply provides validation that prime numbers are different from numbers that are multiples.
— location 1779 Kindle edition

I discussed this criticism with a Maths colleague who disagreed that the constructivist approach would necessarily generate false inferences — but more on that in a later post.

In summary, I am fascinated by the potential of Englemann’s and Carnine’s approach and intend to post more as I mull over its details and implications. Lord help me, but I could not help but be stirred by what could be interpreted as a call to arms:

[Our system] could certainly be improved by a concerted effort to do so. What it needs is a comprehensive critique by serious logicians and philosophers. It needs attention to its details so they can be refined or replaced to be more in accord with logic or empirical evidence.
— location 2591 Kindle edition

And perhaps more importantly, by working teachers too.

(Part Two here)