Applicable Engelmann – The Five Principles for Examples and Nonexamples

This Saturday 13th April 2019 a memorial will be held for the late Siegfried “Zig” Engelmann who died in February of this year (2019).

For those unfamiliar with Engelmann, he is the creator of the teaching method known as Direct Instruction (uppercase D.I). He engineered this teaching method over many years – a theoretical framework for which programs were developed to assist young children learn how to read, spell, solve maths problems, as well as more recently the social sciences.

During Project Follow Through in the 1960s, the largest longitudinal education study of it’s time, his teaching method far out performed other forms on instruction/methods of teaching in three key areas: basic academic skills, problem solving and self-esteem.


There are many places you can read abut Direct Instruction, including an article I recently wrote for Teach Secondary magazine.

What I am interested in is how a teacher take some of the principles of this framework and quickly use them in their own planning and instruction.

A common misconception with Direct Instruction – a scripted form of instruction – is that it only teaches specific cases and, therefore, restricts the universe of concepts that can be taught. In this blog I will describe how one might use the five principles for clear communication through use of examples and non-examples, so that students can then better generalize. A desirable product of the programs is that students are able to transfer what they are taught from the specific cases to more general cases. The ordering and sequencing of such examples is really important.

  1. The wording principle

“To make the sequence as clear as possible, we should use the same wording on all items (or wording that is as similar as possible). This wording helps focus students’ attention on the details of the examples by reducing distraction or confusion that may be caused by variations in teacher language.”


In figure 2.1 you can see an example and non-example of using the wording principle. The left hand example follows the wording principle since the teacher uses minimally different language to describe the concept. The right hand example does not follow the wording example since the teacher uses different language between the examples unnecessarily. The teacher adds excessive variation in language that may cause confusion that would otherwise be avoided following the wording principle.



2. The setup principle

“Examples and nonexamples selected for the initial teaching of a concept should share the greatest possible number of irrelevant features.”

Ambiguity in communication is considered poor in the Direct Instruction model. There are many features that two concepts could share so when we are deciding which to use, it is important that as many of these match up as possible, except the one we want to draw attention to.


Consider that I was teaching you what “on” meant for the first time and showed you this image. This is considered a positive example as it shows us a rectangle lying “on” a line.

Now consider all of the ways that you could describe that image. Someone with no knowledge of this might infer that “on” means something grey, with 4 corners, a 2d shape, a rectangle, an object with a different colour to it’s background etc.

It is only by showing a nonexample that we begin to understand what is meant by on. What is important here, however, is that the nonexample shares all of the irrelevant properties as the positive example – the shape, size, colour – but one – the fact that it is not on the line. By keeping the irrelevant properties invariant, we draw attention to that which is varying – whether the shape is on or not on.

Figure 2.2 shows us what can happen on the right hand side when we don’t follow the setup principle. Students would not be able to infer which aspect of the variation mean the shape is no longer on, since too much is varying at once including the shape itself.

Applying this to my teaching: Consider showing students two examples of how to find the median from a cumulative frequency graph. In example one maximum value on the y-axis in my example was the same as the cumulative frequency of the data. In example 2, I would want to keep everything invariant except that this time the cumulative frequency would be less than the maximum y-axis value. I would be drawing attention to the fact that we need to half the cumulative frequency, not the maximum y-axis value, to read the median.


3. The difference principle

“In order to illustrate the limits or boundaries of a concept, we should show examples and nonexamples that are similar to one another except in the critical feature and indicate that they are different. The difference principle is most effective when the items are juxtaposed— that is, they are shown next to each other or consecutively in a series—making the similarities and differences most obvious.”

To put this simply, we need to choose “nearly examples” (boundary examples) – those that are almost examples but don’t quite make the cut.


Figure 2.3 shows that to demonstrate what horizontal means by example, we should show a positive example of a horizontal object and then show a line that is almost horizontal and provide clarification that this is “not horizontal.” The example on the right hand side shows a positive example and a negative example but the negative example is so far removed from the positive example that it does not show the bound of what constitutes a horizontal object.

Applying this to my teaching: I recently used this when teaching similarity of shapes. Showing two shapes that were similar by using the same scale factor – “these shapes are similar because all of the lengths are being multiplied by the same scale factor” and then showing two of the same shapes where one length had been multiplied by the one scale factor and the other hadn’t – “these are not similar because the lengths are not being multiplied by the same scale factor.” I also showed an example where the same addition had been applied to all lengths – “these are not similar because all of the lengths are not being been multiplied by the same scale factor.” This actually follows principle one, the wording principle, too. I am using the same language for examples and nonexamples so that I am not confusing students by varying my language as well as the examples.


4. The sameness principle

“To show the range of variation of the concept, we should juxtapose examples of the concept that differ from one another as much as possible yet still illustrate the concept and indicate that they are the same. This sequence is intended to foster generalization to unfamiliar concept examples that fall within the demonstrated range.

We are pattern sniffers. A student at any one time could develop a generalization from one example and then apply it incorrectly. The sameness principle shows us that there can be multiple representations of the same concept.

Take the concept of fruit. If I showed you an apple, a pear and a kiwi, one might generalise that all fruit is small and rounded. If I showed you a banana, melon and raspberry then you would have a greater range of the concept of fruit imagery. You would not understand WHY these were fruits yet but at least my choice of examples widened the universe of examples you now have.

Applying this to my teaching: I have applied this to thinking about the wording or commands of a question in my examples. Consider expanding brackets. In my early career, I would have almost certainly showed the students this example:

Expand  2y(3y + 4) and demonstrated how to carry out the procedure.

Using the sameness principle we can demonstrate how the same procedure can be requested using different language:

Expand  2y(3y + 4)

Multiply out 2y(3y + 4)

Write as a power of y    2y(3y + 4)

Write in the form ay^2 + by where a and b are integers to be determined.

We could also easily do this with, for example, identifying angles that are the same in isosceles triangles by keeping the triangle the same size and showing it in different orientations.

5. The testing principle

“To test for acquisition, we should juxtapose new, untaught examples and nonexamples in random order.”

This is blindingly obvious but when creating a mini-quiz or questioning a class, setting up the order of true and not true examples is important so that you can be sure that students are getting questions correct because of their knowledge of that concept, not because they are following a pattern.


Figure 2.5 illustrates this well. The example on the left hand side follows the testing principle since the answers are in a random, unpredictable order. The example on the right does not follow the testing principle since the answers could be answered by following logical predictions based on a pattern. It is important for me that every question I ask in the class room provides me with information that is useful to me so that I can adapt my teaching accordingly. It must be reliable.

The image above is taken from Craig Barton’s and is an activity I designed to try to elicit whether pupils can identify the mode from a frequency table using these principles.

A useful exercise in your department or with a planning buddy or when helping NQTs or non-specialists is to think about:

  1. Examples and boundary (nearly) examples you might use
  2. Which examples might show the greatest range of the concept
  3. The precise language (a mini script) one might use to describe the positive and negative versions of the concept to really lock down concise explanation.
  4. The different ways the same concept could be tested (different command words or wording of questions testing the same skill)
  5. The choice and sequence of questions you ask to test the concept on a formative assessment level.

“Kids never lie. If the responses from students don’t feel like you are being hit by a brick, then you are only presenting, not teaching.” Siegfried Engelmann

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