Insights from DI part 6: Five Principles for Sequencing and Ordering Examples.

This is the sixth post looking at how ideas from Engelmann’s DI can be applied to the everyday classroom. The first five can be found here: one, two, three, four, five

Like the last post, this one will examine The Components of Direct Instruction by Cathy L. Watkins and Timothy A. Slocum, an article from The Journal Of Direct Instruction and an extract from Introduction to Direct Instruction. The paper can be found here.

Direct Instruction teaches ‘generalizable strategies that students can use to solve a wide range of problems’. Instead of teaching a ‘set of discrete specific cases’, it teaches the ‘general case’ and the teaching ‘clearly communicates one and only one meaning and enables students to exhibit generalized responding.’

The authors of the article explain that ‘In order to teach a general case, it is necessary to show students a set of items that includes examples and non-examples arranged so that similarities and differences are readily apparent. Irrelevant aspects of the teaching must be held constant to minimize confusion, and relevant aspects must be carefully manipulated to demonstrate important differences.’

Although the theory behind the sequencing and ordering of examples is incredibly detailed and varies according to the nature of the concept being taught, there are five overarching, general principles that should be followed in order to ensure that communication is clear.

1) The Wording Principle

The idea of ‘faultless communication’ is a key tenet of DI and I touched on this idea here. The wording principle dictates that we should use the same wording for all items in a sequence and, by precluding variance in the language used, we can minimise potential confusion and unnecessary distraction for students. While tiny or even substantial differences in wording may seem trivial to the teacher, they may have catastrophic effects for novice learners. In DI sequences, teacher scripts detail exactly what is communicated, ensuring that this principle is adhered to. This table demonstrates how it is applied in maths.

Scripted lessons are controversial: many people abhor the idea, perceiving them to be dehumanising, dystopian and robotic. This, however, is a clear straw man. If you accept the idea that large parts of the act of teaching involve communication between a teacher and students, then it is self-evident that the clarity of this communication will either hinder or enable learning. Although I am not advocating creating scripted lessons for all aspects of teaching, there are clear benefits to creating communication sequences that are as unambiguous as possible.

How might this apply to the everyday classroom?

a) It may be worthwhile agreeing upon and standardising definitions for different concepts within a department.

b) When teaching concepts, try writing down what you expect to say to see if you are being consistent with the words that you use. Are you using several synonyms for one concept? Are you explaining an idea using unnecessarily technical language? Is all of the communication necessary? Is your explanation meandering and protracted?

c) Try to ensure that communications are not contradicted by later examples. If you define a verb as ‘an action word’, then this is problematic because many verbs have more than one word ( I am eating toast/The boys have been watching the news). If you are interested in the idea of creating communication sequences that avoid complication, ambiguity and contradiction, The Rubric for Identifying Authentic Direct Instruction Programs provides examples of flawed sequences of communication and suggestions for how they could be improved.

2) The Setup Principle

According to this principle, ‘Examples and non-examples selected for initial teaching of a concept should share the greatest possible number of irrelevant features’. This means that examples and non-examples should vary in only one way with all other aspects and features held constant. By doing this, you create a situation where interpretations and inferences are controlled, ensuring that ‘only one interpretation is possible’. Figure 2.2 demonstrates this idea:

The items in the left column differ in only one way, meaning that a student could only logically infer one meaning of ‘on’. On the right, there are numerous variables between the example and the non-example. A novice learner could logically infer that ‘on’ means any of these things:

a) ‘on’ means rectangular

b) ‘on’ means things with corners

c) ‘on’ means horizontal

d) ‘on’ means light grey.

Because a student could make any of these inferences, the setup principle has been violated and the presentation would be considered unnecessarily ambiguous.

In Theory of Instruction, Engelmann talks about ‘stipulation’. This is the idea that ‘the presentation implies that all features of these examples are necessary to the label. The result is that if the learner is presented with variations in any features, the learner will not treat the example in the same way’. After studying multiple examples that follow the left column of Figure 2.2, there may be a danger that a student will infer that ‘on’ only refers to rectangular objects. To prevent this, subsequent sequences would follow where the setup is changed, perhaps by using different shapes or objects or surfaces, demonstrating that ‘on’ is a wide ranging concept.

How might this apply to teaching English?

I have begun experimenting with creating sequences of examples and non-examples in order to teach grammar and specific sentence constructions. Students seem to be responding well to this approach, although I am certain that there are numerous tweaks and improvements that need to be made if the sequences are to fully conform to Engelmann’s rigorous theory.

Here are some examples from teaching present participles:

The first column contains only one variable between the example and non-example, meaning that a learner can only make one logical inference about the meaning of ‘present participle.’ In the second column, there are numerous variables between the example and non-example. Although there may be many more inferences, a learner could logically infer that ‘present participle’ means:

a) The inclusion of the word ‘Enfield’.

b) A sentence that is in the present tense.

c) A sentence that is in the active voice.

3) The Difference Principle

Carefully choosing non-examples is a crucial factor in helping students understand the ‘limits or boundaries of a concept’. To understand what something is, it is helpful to comprehend what it is not. Figure 2.3 demonstrates this idea:

The column on the left provides far more accurate and precise information as to the ‘point at which an example is no longer horizontal’ because both example and non-example are highly similar. The difference in orientation is only several degrees. On the right, the examples do not provide clear information as to the delineation between horizontal and not-horizontal, the difference of orientation spanning 90 degrees. For the difference principle to be most effective, examples and non-examples should be juxtaposed consecutively, making ‘the similarities and differences most obvious.’

How might this apply to the teaching English?

Here are some examples from teaching participial phrases:In the left hand column, the non-example is a gerund phrase (the subject of the verb ‘was’). Gerund phrases are often confused with participle phrases and the juxtaposition of these two examples demonstrates why that is: they are incredibly similar. The non-example here is helpful as it gives precise information as to the delineation between ‘present participle’ and ‘not present participle’. In the right hand column, the non-example is massively different, making it harder for a student to ascertain the boundaries of the concept being taught.

Like with the setup principle, to avoid stipulation (the idea where a student thinks that the examples in a sequence encapsulate the full range of the concept and that other, different examples will therefore fall outside of it), subsequent sequences would follow where the examples are changed, perhaps by using different sentence constructions or subject matter, demonstrating that ‘present participle’ is a wide ranging concept.

4) The Sameness Principle

In order to demonstrate the range and scope of a concept, we should juxtapose maximally different examples. If we were trying to teach the concept of ‘dog’, then we should choose examples that represent the widest possible variety of dogs. Let’s look at an example:

Although we could debate whether there are more strikingly different examples of dogs that we could use, these three have been chosen because, despite all being dogs, they are massively different. If we had merely shown different breeds of terrier, then a student may infer that any future examples that are not terriers would fall outside of the concept of ‘dog’: again, this is what Engelmann refers to as ‘stipulation’.

How does this apply to teaching English?

Here are some example from teaching participial phrases:

The right hand column only demonstrates a tiny range of possible examples of the concept. If we had merely shown these examples, then a student may infer that any differing future examples fall outside of the concept of ‘present participle’. They may logically infer that present participle sentences:

a) Always contain two words.

b) Always begin with a word that ends in ‘ing.’

c) Always precede the subject of a sentence.

d) Always begin a sentence.

d) Always contain the word ‘gossip’.

The left hand column demonstrates a far wider range of examples. I have deliberately inserted an example that includes a quotation as this is how students will most frequently apply these constructions. The second example begins with an adverb, preventing students from inferring the misrule that all present participle phrases begin with an ‘ing’ word. The third example has the phrase at the end of a sentence, demonstrating that these constructions are not always used at the beginning. A full sequence would contain many more maximally different examples, further broadening the scope of the concept.

5) The Testing Principle

After demonstrating several examples and non-examples, ‘to test for acquisition, we should juxtapose new, untaught examples and non-examples in random order’.  Figure 2.5 demonstrates this idea:

In order to ensure that that we are receive accurate information about a student’s understanding of a concept, we need to create tests that do not follow a predictable pattern.

All of these five principles are presented here as separate guidelines. Creating sequences of communication through examples and non-examples often requires multiple sets of juxtaposed examples and non-examples, which helps to avoid misrules and faulty inferences, ensuring that students not only learn the precise point when something stops being a concept (like the horizontal line and the line that is ever so slightly slanted), but also that they learn that a concept can contain innumerable varieties and differences yet still have the same label (like a Chihuahua and an Irish wolfhound: massively different, but still dogs.)

Next post: Insights from DI part 7: Instructional Formats

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