Principles of Engelmann's Direct Instruction: Defining range/scope

This post is part of a series where I go in to detail on some of the main aspects of Direct Instruction as laid out by Siegfried Engelmann in this book.

Below are links to the other posts in this series. Scroll past them to read the article.

• Atomisation
• Overtisation
• Expansion and Context Shaping
• Cognitive load
• Review
• Lesson structure and schemes of work
• Speed principle
• Difficulty and Motivation
• Defining range/scope
• Categories for different types of knowledge
• Instruction for basic types of knowledge
• Instruction for linked types of knowledge
• Instruction for routines
• Instruction for problem solving techniques
• "Real world" maths
• Prompts and scaffolding
• Correcting mistakes
• My take on the strengths and weaknesses of Direct Instruction

Deciding on the range

Once you have atomised your curriculum in to components, you need to thing about the range of all possible examples of each of them. When doing this, you should think about how students will use this skill in the long term, not just about the current topic.

For example, if you were teaching "Solving one-step equations using opposite operations", you would need to include equations like:

 and 

as well as x, ÷, + and -.

It is important to note that this does not mean that these all must be taught straight away. Instead, you can show these as examples of the topic that "we haven't learned how to do yet". Then you can teach these sub-components at a separate date before blending them in to the other example types.

There are several advantages to this approach:

• It effectively pre-teaches elements of several topics that will later build upon the component (in this example, trigonometry, formulae, solving quadratic equations, etc.).
• It highlights the links between these topics by teaching the components of the topics that require the same response as a single component.
• It allows students to see the full scope of the new skill and its potential applications. This also demonstrates the value of learning the skill.
• By showing the full range and limitations early, you are avoiding stipulation where students under-generalise or over-generalise the skill or pattern.

Demonstrating range efficiently

This process often involves vastly broadening what may previously have been a simple component. In order to maintain a high pace on your activities, it is important to use an efficient method to demonstrate the full range of a component and its limitations. In DI, this is achieved by juxtaposing examples using the sameness principle and the difference principle.

Difference principle

In order to demonstrate the full range of applications of a component, you should juxtapose (put next to) examples and questions that are as different as possible (in a single dimension). This allows students to interpolate all the questions in between that would also work.

For example:

From example 1-2 and 2-3, only one dimension has changed (the number on the right-hand side), but it has changed in drastic ways:

• Between 1-2, the answers turn from natural numbers to negative decimals.
• Between 2-3, students must now divide a decimal by 5.

Between 3-4, a new dimension changes (the co-efficient of x becomes a power).

Through the use of four examples, we have showed how the same technique works in several different situations, and allowed students to infer how to apply the method to many questions in between.

Sameness principle

In order to demonstrate the limitations of a component, you should juxtapose pairs of positive/negative examples that are as similar to each other as possible. This allows students to see the exact boundary of where the component stops working and extrapolate to other questions that this skill will not work for. Also, because "any sameness shared by both positive and negative examples rules out a possible interpretation", every dimension that has not changed between the examples cannot be the cause of whether a technique works or does not. 

For example:

Each question now looks highly similar, with many features remaining the same, but 2 can not be solved as a one-step equation. From this students will quickly infer that only equations with single variables can be solved using this technique.

By applying the sameness and difference principles, you can effectively and efficiently show the full range and limitations of a skill.