Viewing what to teach through concepts and operations

The following information is from Unit 18 of the book Teaching: A Course in Psychology by Wesley C. Becker, Siegfried Engelmann, and Don R. Thomas. 

Note: Science Research Associates published the book in 1971. I am unsure if the publishers reprinted the book or if it is available to buy. I found it through a university inter-library loan program. I suspect that Becker or Engelmann use the book or some form of it in their special education teaching classes at the University of Oregon.

• Concept teaching
• Terms
• Concept: idea you want to teach
• Example: you want a student to say that  some red things are red
• Teaching set: the examples you use to teach
• Example: the red things that you present
• Concept instances set: all things related to the concept
• Example: all possible red things
• Non-examples: things unrelated to the concept to discriminate
• Example: other colored things
• A concept is taught when any or all members of the concept set are correctly identified
• Even if some members were not in the teaching set
• Non-examples are responded to differently (not identified)

• Behavioral view of teaching
• Stimulus control of behavior: how stimulus events control a response
• In the presence of examples of the concepts, reinforce one response
• Example: reinforce a student pointing to a red thing when you ask, “Is it red?”
• In the presence of non-examples of the concepts, reinforce another response
• This can also be NO response
• Example: reinforce a student for NOT pointing or for shaking his head “no” when you show a blue thing
• Teach the student more and more critical concept characteristics to insure a concept is taught
• Example of when critical concepts characteristics are not taught
• A student can identify a square by saying its name
• The teacher has shown him a small black square and a big white circle
• The teacher shows him a small black circle (he says “square”)
• The teacher shows him a small white box (he says “not square”)
• The teacher shows him a big black circle (he says “square”)
• The teacher shows him a bigger black circle (he says “square”)
• The student discriminates the shapes on the colors (all black things are squares)
• What to do:
• Pay attention to teach the concept based on the shape and present a variety of colors and sizes
• OR teach both the circle and square at the same time with the exact same size and color
• Further example of misunderstanding of concept characteristics
• The teacher presents the circle and square at the same time with the exact same size and color (white and medium size)
• The student correctly says “square” and “not square”
• The teacher shows him a medium white rectangle (he says “square”)
• The teacher shows him a medium white triangle (he says “square”)
• The teacher shows him a medium white circle (he says “not square”)
• The student knows to look for corners or straight sides, but not for FOUR EQUAL ANGLES and FOUR EQUAL SIDES
• Present more non-examples as the student internalizes the the concept
• Example: present a rectangle, triangle, and circle while teaching the square
• Teach only the essential characteristics of a concept
• Rules
• It is not possible to teach a concept through one example and one non-example
• You must show a concept instances set and non-examples
• The concept instances set must have all essential concept characteristics
• The non-examples must lack all or some of the essential concept characteristics
• Vary stimulus characteristics that are not essential to the concept set and non-examples
• Concept learning happens through two discriminations
• Discriminate relevant characteristics of the concept set from relevant characteristics of the non-examples
• Discriminate irrelevant characteristics in the concept set and non-examples
• Example of teaching color
• Red cannot be taught without also showing non-examples of red such as blue, orange, green
• You are not teaching to discriminate blue from orange at this time
• Red needs to be taught by teaching that texture, pattern, shape, brightness, position, or size of things are all irrelevant

• Engelmann’s definition of concept from his book Conceptual Learning
• Uses all three concepts of a functional definition of behavior (stimulus event, procedure, and behavioral response)
• A concept: set of characteristics shared by a group of instances and not shared by other instances in a universe of concepts
• Terms
• Shared characteristics: physically definable characteristics
• Instances: basic units to give examples of a concept
• Universe of concepts: the set of concepts that the concept comes from and will be discriminated from
• Procedural and functional steps
• Identify a stimulus event that shares specified physical characteristics
• Reinforce one response for this stimulus event
• Reinforce another response for anything other than the stimulus event
• To test a concept
• Present new members of the concept class
• Check for correct responses (behaviors)
• Look at sets of concepts and common characteristics to design teaching strategies
• Essential characteristics of a concept change as other concepts included in the universe of concepts change
• Example: essential characteristics of a dog are paws, small size, light weight, and bark
• If you introduce a horse (non-example of dog) its essential characteristics are hooves, big size, heavy weight, and whinny
• If you introduce a mouse (non-example of dog) its essential characteristics  are claws, small size, light weight, and squeaks
• Now you have to refine the characteristics for small size and light weight for dogs
• You specify the essential discriminations that you teach to avoid confusing the concept instances with non-examples
• Concept analysis
• Determine the essential characteristics of the concept set you want to teach
• Known as S+ (S for stimulus)
• Determine the essential characteristics that make non-examples of the concept set different
• Known as S-
• Determine irrelevant characteristics of the concept set and/or the non-examples
• Known as Si
• Example: “squareness”
• Teach squareness in a universe of concepts with closed geometric figures (like circles, triangles, polygons, etc.)
• Essential characteristics of the concept set: closed figures with 90 degree angles, equal length sides, and straight lines (S+)
• Non-examples: closed figures with any other angles, or unequal sides, or curved lines, or irregular lines (S-)
• Irrelevant characteristics for both groups: size, color, etc. (Si)
• To teach squareness you teach S+ from S- and Si
• Also teach S- from Si
• Examples of pictures to use
• Squares with different colors and sizes and positions
• All other shapes with white color only and different positions
• How to design programs
• Use the essential characteristics to teach discriminations
• Use knowledge of some discriminations to move onto new ones
• Focus on teaching concepts relevant to what the student already knows
• Example with saying sounds for letters
• The universe of concepts is letters and sounds
• The first concept is the letter and sound “m”
• The non-example is a picture of anything (such as a fish)
• Introduce more letters and sounds
• Start with letters and sounds that are least similar (easiest to discriminate)
• Like “m” and “a”
• But “m” and “n” or “d” and “b” are NOT similar
• Avoid similar letters and sounds for later when the student knows more concepts

• Concepts and operations
• Operations or generalized response classes: behavior that functions outside of where it was taught
• Concepts: essential stimulus characteristics that are shared by a set of instances and not shared by non-examples in a universe of concepts
• Operations: essential response characteristics shared by a set of response in a given universe of operations
• Instances are the same for concepts as individual responses are for operations
• Examples
• Concept of color instances: blue bucket, red paper, green plant
• Concept of placing instances: place the box on a table, place the clothes in the dresser, place the paper on the desk
• A program teaches concepts through a set of instances
• The program teaches the range of instances, the range of non-examples, and the range of irrelevant stimulus characteristics
• A program teaches operations through a set of tasks
• The program teaches the essential characteristics of the operation, the essential characteristics of non-examples, and irrelevant response characteristics
• Reinforce correctly performed tasks
• Teaching both concepts and operations at the same time is possible
• Example: move something in relation to something else
• You can make unlimited operations such as place the box on a table, throw the clothes in the dresser, lift the paper on the desk
• An operation can become a concept
• Your observation of a student performing a learned operation is your concept of that operation

• Example of teaching imitation
• A student was in a class for low-functioning students and did not know how to talk or imitate others
• The signal the teacher chose to teach imitation was the words “do this” and then follow the words with the action to imitate
• The first task was for the student to raise her arm
• The teacher said “do this” and raised her arm
• The student did nothing
• So the teacher raised her arm for her (passive shaping)
• After several responses the student raised her arm on her own
• The second task was for the student to tap the table
• The teacher said “do this” and tapped the table
• The student raised her hand
• To discriminate between the two operations, the teacher took her hand and tapped the table for her
• After several responses the student tapped the table on her own
• Then the teacher went back and forth between the operations
• To teach that “do this” (imitating) involves looking at what the teacher does and then perform it
• New tasks were introduced as the student imitated the previously learned tasks accurately
• She learned the new tasks faster and faster
• After 50 tasks, the student imitated over half the new tasks the first time they were shown to her
• She learned the operation of imitation
• The student also learned how to imitate chains of responses
• Such as “do this,” then stand up, walk across the room, and clap your hands
• The teacher also started teaching the student to imitate sounds of words by inserting the sounds in the middle of a chain the physical actions
• Such as “do this,” then stand up, turn around, say “ah”, and sit down

• Operations needed for school success
• Imitation
• Follow instructions
• Pay attention
• Respond on signal
• Work for periods of time
• Persistence
• These operations must be taught if the student does not demonstrate them

• Habits and operations
• Concept teaching and operation teaching can be taught in a classroom but applied anywhere else
• Example: Verbal instructions to “turn left” can be taught in school but later used to learn how to drive a car
• Habits are done efficiently when they are done the same way each time
• The problem with habits is when the conditions change for the habits and they must be unlearned for new ones to be taught

• Example of using teaching concepts and operations to teach single-digit addition without regrouping
• Steps
• The student is taught to count up to a number (rote chain, like 1, 2, 3, 4 to 10)
• The student is taught to count objects up to that number
• Counting objects is an operation to determine how many there are
• The student is taught to identify symbols (numbers) that represent the counting
• The student is taught to make pictures of the symbols
• The student is taught that plus (+) means more
• The student is taught that equals (=) means what is on the left side of the sign must be on the right side of the sign
• 4+5= ? can be solved by drawing 5 pictures under the 5 and counting, “4, 5, 6, 7, 8, 9”
• The student is taught to use his fingers to solve the problems instead of drawing pictures
• The student is taught to count from one number to another to solve problems like 4+?=9 or 9=4+?
• What is good about this system is that after just a little teaching of the operations, the student can do solve many sets based on the operations
• Also the student learns algebraic equations concepts

• Example of using teaching concepts and operations to teach reading by learning ten sounds
• Steps
• Teach the ten sounds
• Teach blending skills at the same time as the sounds
• Now the student can potentially read 720 three sound words, 4320 four sound words, and 21,6000 five sound words!
• This is permutation of words (some are nonsense words but would give practice of the sounds)
• Teach all forty sounds of English
• Now the student can read much of the words in our language
• Teach irregular words at the same time as decodable words
• Operations for reading
• Blending (example: “maaaaaaaaaan”)
• Saying the word as a whole unit (example: “man”)
• Changing a sound in the word (example: “map” or “pan” or “men”)