In this post, I aim to look at the similarities and differences between the sequences and programs as described by Engelmann and Carnine in Theory of Instruction compared to the sequences and exercises described as variation theory. This post is in no means meant to claim that one is better than the other, only to show the similarities and differences between the two to prompt further thinking and discussion.
This post is broken down into parts:
Part 1: Examples and Non-Examples or Positive and Negative Examples
Part 2: Minimally Different Questions for Transformations and Correlated Features
Part 3: A Short aside about Essential and non-essential features
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Printable version here. It has much better formatting and maths text than Weebly will handle.
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This post is broken down into parts:
Part 1: Examples and Non-Examples or Positive and Negative Examples
Part 2: Minimally Different Questions for Transformations and Correlated Features
Part 3: A Short aside about Essential and non-essential features
------------------
Printable version here. It has much better formatting and maths text than Weebly will handle.
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Introduction
A fish does not know the concept of "water" until it is removed from the lake it has always lived to experience not-water. You cannot understand what a triangle is by only looking at triangles (or worse very specific, common triangles).
Variation theory often looks at the idea of making differences more discernible by keeping features the same whether by showing examples and non-examples of a concept, or questions where minimal changes occur between them. Ference Marton describes the ‘dimensions of possible variation’ and ‘range of permissible change’ for a given concept. That is, what features can be changed and how far can you change them while still being an example of the concept. Anne Watson adds that it is the variation against a backdrop of invariance that allows the features to be discerned.
Engelmann also uses these ideas in Theory of Instruction. Two of his five structural requirements for generalisation are that the positive examples of the concept must be distinguished by one and only one quality, and that the examples must demonstrate the range of variation to which the learner will be expected to generalise.
A key difference between variation theory and Theory of Instruction that I've noticed, from my reading, is that variation theory is very broad about what it is applied to and how it is applied. It is one of the reasons I'm always ending up on the fence about it, it's never felt specific enough to apply consistently well. (If there is more specific details than considering the dimensions and range of variation, please let me know where I can read more.)
Theory of Instruction on the other hand is very specific about how it approaches variation for different types of concepts and knowledge. There are a number of sequences that can be used depending on what is being taught. Some sequences are more appropriate for a given idea than another.
For instance, as I read Theory of Instruction, I kept trying to work out where classifying surds would come under as every other sequence seemed to apply to it. Surds could be taught through a non-comparative sequence, a noun sequence, a single-transformation sequence, a correlated-features sequence, or a correlated-features transformation sequence. And that isn't even touching the programs! Each type of sequence would correspond with a different type of learning of understanding what is and isn't a surd. The way that you approach a topic/concept/fact changes the sequence you would use and consequently what the students will learn from that sequence.
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Part 1: Examples and Non-Examples or Positive and Negative Examples
First, I must clarify some terminology as Engelmann describes examples differently to what is now common place. Engelmann will give an example that may or may not be an example of the concept under consideration. For instance, the example could be a word, a potato, an equation, etc. It is just that, an example. If the example is an example of the concept, he calls it a positive example, what we would generally call an example. If the example is not an example of the concept it is a negative example, what we would now generally call a non-example. To improve clarity in a number of paragraphs I've used "positive example" and "negative example" rather than "example" and "non-example". In short, positive example = example, and negative example = non-example.
Both variation theory and Engelmann agree that the boundaries of what something is and what something is not must be shown and understood. That is, showing examples, non-examples, and almost examples. Engelmann has a few extra principles we can keep in mind when deciding on examples and non-examples.
Some of Engelmann's Principles
Engelmann's difference principle means that if we want to show that two successive examples are different (example and a non-example), we should use a minimally different example. Change the smallest possible thing to make it change between a positive and negative example. Engelmann's sameness principle goes the other the other direction, if we want to show that two successive examples are the same (two examples or two non-examples), we should use a maximally different example. Change the example as much as possible and so that it is still a positive example.
A fish does not know the concept of "water" until it is removed from the lake it has always lived to experience not-water. You cannot understand what a triangle is by only looking at triangles (or worse very specific, common triangles).
Variation theory often looks at the idea of making differences more discernible by keeping features the same whether by showing examples and non-examples of a concept, or questions where minimal changes occur between them. Ference Marton describes the ‘dimensions of possible variation’ and ‘range of permissible change’ for a given concept. That is, what features can be changed and how far can you change them while still being an example of the concept. Anne Watson adds that it is the variation against a backdrop of invariance that allows the features to be discerned.
Engelmann also uses these ideas in Theory of Instruction. Two of his five structural requirements for generalisation are that the positive examples of the concept must be distinguished by one and only one quality, and that the examples must demonstrate the range of variation to which the learner will be expected to generalise.
A key difference between variation theory and Theory of Instruction that I've noticed, from my reading, is that variation theory is very broad about what it is applied to and how it is applied. It is one of the reasons I'm always ending up on the fence about it, it's never felt specific enough to apply consistently well. (If there is more specific details than considering the dimensions and range of variation, please let me know where I can read more.)
Theory of Instruction on the other hand is very specific about how it approaches variation for different types of concepts and knowledge. There are a number of sequences that can be used depending on what is being taught. Some sequences are more appropriate for a given idea than another.
For instance, as I read Theory of Instruction, I kept trying to work out where classifying surds would come under as every other sequence seemed to apply to it. Surds could be taught through a non-comparative sequence, a noun sequence, a single-transformation sequence, a correlated-features sequence, or a correlated-features transformation sequence. And that isn't even touching the programs! Each type of sequence would correspond with a different type of learning of understanding what is and isn't a surd. The way that you approach a topic/concept/fact changes the sequence you would use and consequently what the students will learn from that sequence.
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Part 1: Examples and Non-Examples or Positive and Negative Examples
First, I must clarify some terminology as Engelmann describes examples differently to what is now common place. Engelmann will give an example that may or may not be an example of the concept under consideration. For instance, the example could be a word, a potato, an equation, etc. It is just that, an example. If the example is an example of the concept, he calls it a positive example, what we would generally call an example. If the example is not an example of the concept it is a negative example, what we would now generally call a non-example. To improve clarity in a number of paragraphs I've used "positive example" and "negative example" rather than "example" and "non-example". In short, positive example = example, and negative example = non-example.
Both variation theory and Engelmann agree that the boundaries of what something is and what something is not must be shown and understood. That is, showing examples, non-examples, and almost examples. Engelmann has a few extra principles we can keep in mind when deciding on examples and non-examples.
Some of Engelmann's Principles
Engelmann's difference principle means that if we want to show that two successive examples are different (example and a non-example), we should use a minimally different example. Change the smallest possible thing to make it change between a positive and negative example. Engelmann's sameness principle goes the other the other direction, if we want to show that two successive examples are the same (two examples or two non-examples), we should use a maximally different example. Change the example as much as possible and so that it is still a positive example.
These two principles capitalise on interpolation and extrapolation. From two maximally different positive examples, we can infer that everything in between is also a positive example. From two minimally different examples, we can infer that anything beyond the negative example, is also a negative example.
Example
If we say that 0.1 is a decimal number and 12125948719.183601312947 is a decimal number, then we can infer that anything in between will also be a decimal number.
Example
If we say that 3.4 is a decimal number but ¾ is not a decimal number, we can see what minimal change it took to get a negative example (decimal point or a fraction bar). Yes, ¾ can be written as a decimal, but it currently is not. The conversion fact would be taught in another sequence (consider the extra cognitive load!).
Engelmann says that when showing an examples we must make clear whether it is a positive example or a negative example. A clear, unambiguous signal should be given for each example. Visually, this may be a tick and cross. Verbally, this may be yes and no, true and false, or "it is X" and "it is not-X" by keeping in mind the wording principle (using the same wording on juxtaposed examples). The benefit of using "it is X" and "it is not-X" is that students must say or write the new vocabulary repeatedly which will help to add the word to their lexicon (by orthographic mapping - 3 times). Whereas yes/no responses do not have them practise saying or writing the new word(s).
If we say that 0.1 is a decimal number and 12125948719.183601312947 is a decimal number, then we can infer that anything in between will also be a decimal number.
Example
If we say that 3.4 is a decimal number but ¾ is not a decimal number, we can see what minimal change it took to get a negative example (decimal point or a fraction bar). Yes, ¾ can be written as a decimal, but it currently is not. The conversion fact would be taught in another sequence (consider the extra cognitive load!).
Engelmann says that when showing an examples we must make clear whether it is a positive example or a negative example. A clear, unambiguous signal should be given for each example. Visually, this may be a tick and cross. Verbally, this may be yes and no, true and false, or "it is X" and "it is not-X" by keeping in mind the wording principle (using the same wording on juxtaposed examples). The benefit of using "it is X" and "it is not-X" is that students must say or write the new vocabulary repeatedly which will help to add the word to their lexicon (by orthographic mapping - 3 times). Whereas yes/no responses do not have them practise saying or writing the new word(s).
Finally, Engelmann's setup principle states that we should use an example and non-example that share the greatest number of common traits as possible, as this will rule out the largest number of not important factors as possible.
Consider the following set of examples and non-examples (read across the rows). One point remains constant throughout. Between a non-example and an example, the smallest (perceptible) change possible is used. Specific examples that students may already know names of are left to the end.
Engelmann's Sequences for Examples and Non-Examples
The main types of concepts that are classifiable as examples and non-examples Engelmann calls non-comparatives (things that can be changed from an example to a non-example by changing only one feature), comparatives (things that must be compared to something else), and nouns (things that can be changed from an example to a non-example by changing more than one feature). Each has a sequence designed to elicit the boundaries of what is and is not an example.
Engelmann suggests two types of sequences for non-comparatives: negative-first sequences and positive-first sequences, named for the first example that we show in the sequence. These sequences apply the difference and sameness principles to get across what something is and what it is not in less than 7 examples
The main types of concepts that are classifiable as examples and non-examples Engelmann calls non-comparatives (things that can be changed from an example to a non-example by changing only one feature), comparatives (things that must be compared to something else), and nouns (things that can be changed from an example to a non-example by changing more than one feature). Each has a sequence designed to elicit the boundaries of what is and is not an example.
Engelmann suggests two types of sequences for non-comparatives: negative-first sequences and positive-first sequences, named for the first example that we show in the sequence. These sequences apply the difference and sameness principles to get across what something is and what it is not in less than 7 examples
Keep in mind, a feature is irrelevant and should not be varied if changing it does not convert a positive example into a negative example. For instance, writing √3, the same but in bold, √3, or the same but in a different colour, √3, cannot change the surd into a not-surd. Being bold or a different colour is not a property of being a surd or a not-surd. Showing a set such as √2 (+), √3 (+) and √4 (-), eliminates bold and colour from being possible means of changing a surd to a not-surd as we can do that without changing those features. However, if we had changed them as well (√3 (+) and √4 (-)) then we cannot decide if it is the fact that it is bold or a different number that has made it a not-surd.
We need to ensure a single interpretation by the seventh example as students have to be able to answer whether the following test items are examples or non-examples in the testing phase. If boundaries are not clear by then, students are likely to be unable to answer some test items correctly. The test segment should repeat some earlier examples, contain new positive and negative examples, contain sufficient positive examples to test understanding, and not be predictable in order of positive and negative.
Non-comparatives are one of the more frequent examples shown in variation theory, often with yes/no responses.
Positive- and negative-first sequences are also used for comparatives. However, we also need to account for how much it changes and whether that remains a positive or negative example. For a range of 1 to 10, where 1 is a low as you can reasonably make it (but not 0) and 10 is as high as you can reasonably make it, Engelmann suggests starting at around a 3 or 4 to establish a base line to compare from, each successive example is compared to the previous example not the original. Positive changes should include a small change (+~1), a quite-large change (+~4), and an intermediate change (+~2). Negative changes should include a no-change (-~0), and minimum change (-~1) that students will be to tell the difference between. Notice, the sequence does not include 0 despite the fact that a comparative could possibly be 0.
We need to ensure a single interpretation by the seventh example as students have to be able to answer whether the following test items are examples or non-examples in the testing phase. If boundaries are not clear by then, students are likely to be unable to answer some test items correctly. The test segment should repeat some earlier examples, contain new positive and negative examples, contain sufficient positive examples to test understanding, and not be predictable in order of positive and negative.
Non-comparatives are one of the more frequent examples shown in variation theory, often with yes/no responses.
Positive- and negative-first sequences are also used for comparatives. However, we also need to account for how much it changes and whether that remains a positive or negative example. For a range of 1 to 10, where 1 is a low as you can reasonably make it (but not 0) and 10 is as high as you can reasonably make it, Engelmann suggests starting at around a 3 or 4 to establish a base line to compare from, each successive example is compared to the previous example not the original. Positive changes should include a small change (+~1), a quite-large change (+~4), and an intermediate change (+~2). Negative changes should include a no-change (-~0), and minimum change (-~1) that students will be to tell the difference between. Notice, the sequence does not include 0 despite the fact that a comparative could possibly be 0.
For example, the concept of steeper could be taught by rotating a flat hand through a range of angles from horizontal to straight up and could start at about a 30° angle. From there, rotations of the appropriate sizes could be applied. By the wording principle student should state if it gets steeper or doesn't get steeper. Using phrases like more steep, same steepness, or less steep is not advised as it is not the focus of the initial sequence: steeper/not-steeper is. A later sequence could introduce that step.
When testing, you should include the full range of 1 to 10, include some large changes, some small changes, and some no-changes, and include at least one value that acts as both a positive example and a negative example. For example, a change from 4 to 5 is positive but then a change from 6 to 5 is a negative. The value of 5 is then not what determines if it is positive or negative, it is the change to that value that is.
I have not seen many variation theory type exercises for comparatives.
Finally, we have nouns. Similar to non-comparatives, a noun either is or is not an example in its own right, as opposed to being compared to something. However, unlike non-comparatives, nouns can be changed in more than one way to create examples and non-examples. That is, the boundaries of what is and is not a particular noun are more hazy.
Noun sequences do not follow the same positive-first or negative-first sequences. Noun sequences always begin with greatly different positive examples to show sameness. The wider the positive range, the more examples needed to show full variation, but keep the sequence as short as possible. Keep in mind that this is for an introductory sequence, so use clear cut examples not debatable ones.
When testing, you should include the full range of 1 to 10, include some large changes, some small changes, and some no-changes, and include at least one value that acts as both a positive example and a negative example. For example, a change from 4 to 5 is positive but then a change from 6 to 5 is a negative. The value of 5 is then not what determines if it is positive or negative, it is the change to that value that is.
I have not seen many variation theory type exercises for comparatives.
Finally, we have nouns. Similar to non-comparatives, a noun either is or is not an example in its own right, as opposed to being compared to something. However, unlike non-comparatives, nouns can be changed in more than one way to create examples and non-examples. That is, the boundaries of what is and is not a particular noun are more hazy.
Noun sequences do not follow the same positive-first or negative-first sequences. Noun sequences always begin with greatly different positive examples to show sameness. The wider the positive range, the more examples needed to show full variation, but keep the sequence as short as possible. Keep in mind that this is for an introductory sequence, so use clear cut examples not debatable ones.
Students are then tested on the examples that have been shown using some variation of the question “What is this?” where they are required to answer using the name of the noun shown not using "X" and "not-X" or other yes/no responses.
Following on from the positive-only response test, the discrimination test works through nouns that are dissimilar, then similar in shape, then similar in name, then similar in name and shape (using examples they already know the names of).
Since we began with positive-only variation, we can eliminate those variables that keep examples positive and use only minimally different non-examples to highlight the differences more clearly (difference principle).
The recent increased usage of examples and non-examples by teachers should lead to an increased student understanding of what something is and isn't. However, using Engelmann's principles and sequences, the effectiveness of them could be improved. Something to keep in mind with his sequences though, is that they never explicitly define what something is. A negative-first sequence involving what is and isn't a triangle never states a definition of a triangle. Students just build up a mental model of what it is and isn't.
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Part 2: Minimally Different Questions for Transformations and Correlated Features
Many of the types of sequences or exercises shown for variation theory are of the form shown below. An exercise where a single idea is applied to minimally different examples through a range of variation. The extent of that range varies quite between sets of questions. In Engelmann's terms, these types of exercises are for transformations or correlated-features.
Following on from the positive-only response test, the discrimination test works through nouns that are dissimilar, then similar in shape, then similar in name, then similar in name and shape (using examples they already know the names of).
Since we began with positive-only variation, we can eliminate those variables that keep examples positive and use only minimally different non-examples to highlight the differences more clearly (difference principle).
The recent increased usage of examples and non-examples by teachers should lead to an increased student understanding of what something is and isn't. However, using Engelmann's principles and sequences, the effectiveness of them could be improved. Something to keep in mind with his sequences though, is that they never explicitly define what something is. A negative-first sequence involving what is and isn't a triangle never states a definition of a triangle. Students just build up a mental model of what it is and isn't.
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Part 2: Minimally Different Questions for Transformations and Correlated Features
Many of the types of sequences or exercises shown for variation theory are of the form shown below. An exercise where a single idea is applied to minimally different examples through a range of variation. The extent of that range varies quite between sets of questions. In Engelmann's terms, these types of exercises are for transformations or correlated-features.
https://variationtheory.com/2020/02/22/division-with-fractions/
Division of fractions can be thought of as an example of a single-transformation, where two fractions transform into their quotient. A transformation can be used with a variety of examples of the same type (division of two fractions), the learner produces different symbolic responses for different examples (quotients are not all the same), and there is a sameness shared by the responses (quotients can be written as fractions).
Unlike using an example-problem pair, Engelmann suggest modelling at least two minimum difference examples in the middle range of extremes. This is followed by a test of minimum difference examples much like the variation theory questions and should include examples that have the same response (same quotient). The minimum difference should be of only one dimension and be of different dimensions throughout (change dividend's numerator, divisor's numerator, dividend's denominator, divisor's denominator, etc.). The difference is that Engelmann suggests that there be up to six minimally different examples tested before moving onto four to twelve greatly different examples that change more than one dimension at a time. This is so that students become familiar with dealing with the full range of question types.
However, Engelmann says that we cannot teach the transformation through a single sequence. But the initial sequence is the most important. It is designed to show as much as possible about the structure of the concept and the sameness across different examples. We must show all subtypes as quickly as possible to avoid stipulation (that the transformation only applies to a narrow range of examples). Engelmann says that only when we show the learner what is the same about all examples, do we teach the concept.
That is to say, sequences like the ones shown above that include more minimal variation could be used later to get at an underlying idea that is not the main idea. For example, after an initial sequence of dividing fractions (that includes examples from the following sequences), subsequent sequences to target ideas such as dividing by a unit fraction, dividing by an integer, dividing by a fraction whose numerator and denominator are factors of the dividend, dividing by a fraction with a common denominator, and so on. (Note this is purely hypothetical and not a tested set of sequences I have tried with students. Division of fractions may also be taught through a cognitive routine rather than a single-transformation.)
Division of fractions can be thought of as an example of a single-transformation, where two fractions transform into their quotient. A transformation can be used with a variety of examples of the same type (division of two fractions), the learner produces different symbolic responses for different examples (quotients are not all the same), and there is a sameness shared by the responses (quotients can be written as fractions).
Unlike using an example-problem pair, Engelmann suggest modelling at least two minimum difference examples in the middle range of extremes. This is followed by a test of minimum difference examples much like the variation theory questions and should include examples that have the same response (same quotient). The minimum difference should be of only one dimension and be of different dimensions throughout (change dividend's numerator, divisor's numerator, dividend's denominator, divisor's denominator, etc.). The difference is that Engelmann suggests that there be up to six minimally different examples tested before moving onto four to twelve greatly different examples that change more than one dimension at a time. This is so that students become familiar with dealing with the full range of question types.
However, Engelmann says that we cannot teach the transformation through a single sequence. But the initial sequence is the most important. It is designed to show as much as possible about the structure of the concept and the sameness across different examples. We must show all subtypes as quickly as possible to avoid stipulation (that the transformation only applies to a narrow range of examples). Engelmann says that only when we show the learner what is the same about all examples, do we teach the concept.
That is to say, sequences like the ones shown above that include more minimal variation could be used later to get at an underlying idea that is not the main idea. For example, after an initial sequence of dividing fractions (that includes examples from the following sequences), subsequent sequences to target ideas such as dividing by a unit fraction, dividing by an integer, dividing by a fraction whose numerator and denominator are factors of the dividend, dividing by a fraction with a common denominator, and so on. (Note this is purely hypothetical and not a tested set of sequences I have tried with students. Division of fractions may also be taught through a cognitive routine rather than a single-transformation.)
Correlated-features are most simply described as if-then fact statements. Correlated-features sequences use positive-first and negative-first sequences as part of their structure. The difference is that, for "if 1, then 2", the examples show positive and negative examples related to fact part 1. Students must then answer two questions: a question that discriminates between if it is fact part 2 or not-fact part 2 given what is shown, and "How do you know?" or "Why?".
Avoid wording the if-then statements as "if not-1, then 2" as there is greater room for error than "if 1, then 2" as positive examples match and negative examples match.
Example
If a number is a surd, then it is irrational.
√3: Is it irrational? Yes. How do you know? Because it is a surd.
√4: Is it irrational? No. How do you know? Because it is not a surd.
Some if-then statements suggest a transformation, that is a unique response to be made not "it is X" or "it is not-X". For example, if we wanted to teach "if , then read the sum of and ", then the students would not respond to with "it is a sum" or "it is not a sum", they would respond with "the sum of three and four". They have given a unique response for this example. For these if-then statements, Engelmann suggests a modified correlated features sequence called a correlated-features transformation sequence. The difference is that the transformation is stated at the beginning of the sequence and we ask for the response not a discrimination.
What do you get if you transform a transformation? A double transformation. The double transformation program takes one transformation and uses it to teach another related transformation. For example, to teach .
The program begins with testing five maximally different examples from the familiar set that they have thoroughly mastered. Following on is the transformed set, where each of the five familiar examples are transformed into their new form starting with the transformed version of the last familiar example. The first two are modelled, the rest are tested. This allows for within-set juxtapositions to take place. That is, what is happening within this new transformation. This is following by a partial cycle, a four-question set that compares the minimally different across-set pairs beginning with the pair of the last example in the transformed set. Finally, the two sets are integrated and tested at random including previous examples and new examples only from the transformed set. There should be no outside prompts to indicate whether the next example will be from the familiar or from the transformed set.
Avoid wording the if-then statements as "if not-1, then 2" as there is greater room for error than "if 1, then 2" as positive examples match and negative examples match.
Example
If a number is a surd, then it is irrational.
√3: Is it irrational? Yes. How do you know? Because it is a surd.
√4: Is it irrational? No. How do you know? Because it is not a surd.
Some if-then statements suggest a transformation, that is a unique response to be made not "it is X" or "it is not-X". For example, if we wanted to teach "if , then read the sum of and ", then the students would not respond to with "it is a sum" or "it is not a sum", they would respond with "the sum of three and four". They have given a unique response for this example. For these if-then statements, Engelmann suggests a modified correlated features sequence called a correlated-features transformation sequence. The difference is that the transformation is stated at the beginning of the sequence and we ask for the response not a discrimination.
What do you get if you transform a transformation? A double transformation. The double transformation program takes one transformation and uses it to teach another related transformation. For example, to teach .
The program begins with testing five maximally different examples from the familiar set that they have thoroughly mastered. Following on is the transformed set, where each of the five familiar examples are transformed into their new form starting with the transformed version of the last familiar example. The first two are modelled, the rest are tested. This allows for within-set juxtapositions to take place. That is, what is happening within this new transformation. This is following by a partial cycle, a four-question set that compares the minimally different across-set pairs beginning with the pair of the last example in the transformed set. Finally, the two sets are integrated and tested at random including previous examples and new examples only from the transformed set. There should be no outside prompts to indicate whether the next example will be from the familiar or from the transformed set.
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Part 3: A Short aside about Essential and Non-Essential Features
We can choose to vary one of two things within a sequence for teaching new complex responses: the essential features (what makes an example or non-example, or different response), or the non-essential features otherwise called the enabling features (the features that occur in every example but not essential to achieving the objective, e.g., the length of the vinculum for a square root or ). If required, we could alternatively remove a component and teach it separately in another context using a removed-component-behaviour program.
If we use gradient for example: m = (y₂-y₁) / (x₂-x₁)
An essential-response-feature program could begin with students substituting the correct values into the formula and then the teacher or a calculator performs the enabling subtractions and division/fraction simplification.
An enabling-response program could begin with the teacher substituting the correct values into the formula and the student then performs the subtractions and division/fraction simplification.
Components that could be removed include integer subtraction, division of integers, simplifying fractions with integer numerator and denominator.
The usage of these three programs will vary depending on where the students are at in their understanding and what the teacher wants them to focus on practising.
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Conclusion
Variation theory and Engelmann's Theory of Instruction both attempt to give the teacher the means of helping students to discern similarities and differences between examples. Where variation theory focuses on the dimensions and range of variation, Engelmann focuses on specific types of concepts and sequences to teach them.
If you want to read more about Engelmann's Theory of Instruction click here to read my summary of the book and click here to access the book itself.
Part 3: A Short aside about Essential and Non-Essential Features
We can choose to vary one of two things within a sequence for teaching new complex responses: the essential features (what makes an example or non-example, or different response), or the non-essential features otherwise called the enabling features (the features that occur in every example but not essential to achieving the objective, e.g., the length of the vinculum for a square root or ). If required, we could alternatively remove a component and teach it separately in another context using a removed-component-behaviour program.
If we use gradient for example: m = (y₂-y₁) / (x₂-x₁)
An essential-response-feature program could begin with students substituting the correct values into the formula and then the teacher or a calculator performs the enabling subtractions and division/fraction simplification.
An enabling-response program could begin with the teacher substituting the correct values into the formula and the student then performs the subtractions and division/fraction simplification.
Components that could be removed include integer subtraction, division of integers, simplifying fractions with integer numerator and denominator.
The usage of these three programs will vary depending on where the students are at in their understanding and what the teacher wants them to focus on practising.
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Conclusion
Variation theory and Engelmann's Theory of Instruction both attempt to give the teacher the means of helping students to discern similarities and differences between examples. Where variation theory focuses on the dimensions and range of variation, Engelmann focuses on specific types of concepts and sequences to teach them.
If you want to read more about Engelmann's Theory of Instruction click here to read my summary of the book and click here to access the book itself.