Tag Archives: Direct Instruction

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?

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.

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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)

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