‘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.
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. 
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. 
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. 
The goal is simply to teach as little as possible to provide thorough coverage of the content.
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. 
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. 
[However each] exception must be taught because if it is ignored, the learner may not learn it. 
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. 
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. 
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. 
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. 
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. 
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. 
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.
 Carnine, D. and Engelmann, S., Theory of Instruction: Principles and Applications (1982), Kindle location 299
 Carnine, D. and Engelmann, S., Could John Stuart Mill Have Saved Our Schools? (2011), Kindle location 610
 2011 loc 640
 2011 loc 678
 2011 loc 671
 2011 loc 648
 2011 loc 671
 2011 loc 685
 1982 loc 1319-1348
 2011 loc 719
Reblogged this on The Echo Chamber.
I do like top and bottom !
I got most of the part 1 from Engleman. The separation between the ‘yes it is” and the “no it isn’t” is so clearly put, and so non-obvious. It was a book, “Counterexamples in Analysis” which reorganised my opinions about examples.
Thank you! I intend to look at all of Mill’s rules of induction in later posts (although Engelmann and Carnine reinvented them in Theory of Instruction, I think they are more clearly expressed in Mill). Who wrote “Counterexamples in Analysis”? I like what I have read of Engelmann so far. Even if one doesn’t accept the whole DI package, his clarity of thought is transformative.
Re Counterexamples in Analysis, Gelbaum. It is an older book, reprinted often.
“By James Arvo on July 22, 2003
It can happen to anybody. There you are, minding your own business, when the though hits you: Does every continuous function have a derivative somewhere? You try to prove that it must. It sure seems like it must. How could it not? Hours slip by, and you’ve made no progress. What do you do? You pick up Gelbaum and Olmsted’s classic “Counterexamples in Analysis”. There on page 38 is an example of a continuous function that has no derivative; none; anywhere. No wonder you couldn’t prove it.”
Heavy stuff, but fun !