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STOP DOING MATH LONG ENOUGH TO LEARN IT

- Principles of Learning
- Delano P. Wegener, Ph.D.
- Spring 2005

Instruction

- Instruction is much more than presentation of

information. - Instruction may include events that are generated

by a page of print, a picture, a television

program, a combination of physical objects,

potentially many other stimuli, as well as

activities of a teacher.

Instruction

- Teaching refers to the activities of the teacher.

Therefore teaching is only one part (I think it

is an important part) of instruction. - Instruction is a deliberately arranged set of

external events designed to support internal

learning processes.

Internal Conditions of Learning

- Because the internal conditions of learning are

beyond our control we will not elaborate beyond

listing them.

Internal Conditions of Learning

- Reception of stimuli by receptors
- Registration of information by sensory registers
- Selective perception for storage in short-term

memory (STM) - Rehearsal to maintain information in STM
- Semantic encoding for storage in long-term memory

(LTM) - Retrieval from LTM to working memory (STM)
- Response generation to effectors
- Performance in the learners environment
- Control of processes through executive strategies

Internal Conditions of Learning

- Cognitive scientists use that information, but we

make no explicit use of the internal conditions

of learning when designing instruction.

Learning

- Some of the very basic facts and theories about

learning will help to understand the guidelines,

presented later, for studying mathematics.

Learning

- In particular, it is helpful to be aware of
- Conditions of Learning,
- Learning Outcomes,
- Domains of Learning Objectives, and
- Classes of Learning Objectives in the Cognitive

Domain.

Learning

- In this seminar we are especially interested in

how these facts and theories about learning

pertain to learning mathematics.

External Conditions of Learning

- Gaining attention
- Informing the learner of the objective
- Stimulating recall of prerequisite learning
- Presenting the stimulus material
- Providing learning guidance
- Eliciting the performance
- Providing feedback about performance correctness
- Assessing the performance
- Enhancing retention and transfer

External Conditions of Learning

- Because the External Conditions of Learning

directly affect instruction and what you must do

to learn, each of these conditions will be

explained.

1. Gaining attention

- Stimulation to gain attention to ensure the

reception of stimuli. - Various kinds of events can be used to gain the

students attention. These events might be as

simple as calling the class to order or as

complex as the mix of sound, pictures, movement,

and light as found in the most sophisticated TV

commercials.

1. Gaining attention

- An appeal to the students interest is frequently

employed as a means of gaining attention. - For adult students we frequently assume they will

themselves provide the stimulation to gain their

attention.

Informing the Learnerof the Objective

- Informing learners of the learning objective

establishes appropriate performance. - The student must know the kind of performance

that is expected as a demonstration that learning

has taken place. - In general it is a mistake to assume the student

will know the objective of the lesson.

3. Stimulating Recall of Prerequisite Learning

- Reminding learners of previously learned content

for retrieval from LTM - Much of learning is the combination of ideas. If

any of the ideas involved have been learned

previously, the student should be reminded of

them so they are retrieved from LTM into STM

where they are available for immediate recall.

3. Stimulating Recall of Prerequisite Learning

- At the time of learning, previously learned ideas

must be readily available. They must therefore be

recalled from LTM (into STM) prior to the time of

learning.

Stimulating Recall of Prerequisite

LearningMathematics

- Suppose we expect the student to learn the
- Fundamental Theorem of Arithmetic
- Any natural number can be expressed as a

product of prime numbers. - If the definitions of natural number, product,

and prime number are not readily available, then

the Fundamental Theorem of Arithmetic will not be

learned.

Stimulating Recall of Prerequisite

LearningMathematics

- It is essential that these definitions be

recalled from LTM to STM. - Assuming that these definitions have previously

been learned, the teacher can insure they are

recalled into STM by simply reminding the student

of those definitions.

Stimulating Recall of Prerequisite

LearningMathematics

- The adult student might be expected to recall

those definitions simply because the appearance

of the words is enough of a reminder.

Presenting the Stimulus MaterialMathematics

- It is important that the proper stimuli be

presented as a part of the instructional events.

- If a mathematical rule is to be learned, then

that rule must be communicated. Such

communication, if printed, may use italics, bold

print, underlining, colors, etc. to emphasize

particular features.

Presenting the Stimulus Material

- When young children are learning concepts or

rules the stimulus material should present

examples prior to presenting a statement of the

concept or rule. - When adults are learning concepts or rules the

stimulus material should present a statement of

the concept or rule followed by examples.

Presenting the Stimulus MaterialMathematics

- Stimulation presentation for the learning of

concepts and rules requires the use of a variety

of examples. - Thus the stimulus presentation for learning the

mathematical concept of linear function will

involve examples of functions like - f(x) 3x, or f(x) 7
- as well as examples like f(x) 3x 7

5. Providing Learning Guidance

- Communications which have the function of

providing learning guidance do not provide the

answer. - They suggest a line of thought which will lead to

appropriate combining of previously learned

concepts allowing the student to learn the

answer.

5. Providing Learning Guidance

- Communications designed to provide learning

guidance should stimulate a direction of thought

which keeps the student on the right track.

Providing Learning GuidanceMathematics

- When presenting an example of solving a linear

equation the instructor does not encourage a

memorized set of steps to arrive at the answer.

- Rather the instructor constantly reminds the

student of the previously learned two operations

which generate an equation equivalent to the

original equation.

Providing Learning GuidanceMathematics

- All communications in this context are designed

to keep the student on track to generate a

sequence of equations, all equivalent to the

original, terminating in a simplest equation.

6. Eliciting the Performance

- Suppose the previous five events have taken

place, enough material has been presented,

sufficient learning guidance has taken place, and

the student indicates/believes he has learned the

concepts. - It is then time for the student to demonstrate

both to himself and the instructor that he has

learned the concept.

6. Eliciting the PerformanceMathematics

- When studying mathematics, the first five

External Conditions of Learning have occurred

only after the student has studied all materials

(text, lecture, etc.) related to a section of the

textbook.

6. Eliciting the PerformanceMathematics

- The student should then turn to the exercises and

demonstrate to himself that the concept has been

learned. - The purpose of homework and or quizzes is to

demonstrate to both the student and the

instructor that the student has indeed learned

the desired concept.

6. Eliciting the PerformanceMathematics

- Notice that working exercises (at this stage) is

to demonstrate that the concept has been learned.

- It (working exercises) is not a device for

learning the concept. - Therefore it is not necessary to work huge

numbers of exercises.

7. Providing Feedback About Performance

Correctness

- The important characteristic of feedback

communication is not its form but its function - Providing information to the student about

the correctness of his/her performance relative

to the Learning Objective.

Providing Feedback About Performance

CorrectnessMathematics

- Mathematics textbooks generally provide very

minimal feedback in the form of answers to the

odd numbered problems.

Providing Feedback About Performance Correctness

Mathematics

- The Learning Objective is hardly ever find the

answer to a problem. - The Learning Objective is to learn to use a

combination of concepts, rules, processes, etc.

to solve particular types of problems. - The feedback in most textbooks is not very useful

and indeed fosters a misunderstanding of the

Learning Objective.

Providing Feedback About Performance Correctness

Mathematics

- Therefore, the mathematics instructor should

provide feedback which addresses the steps and

the reasons for the steps used by the student

when solving a problem. - That is, the mathematics instructor should

provide feedback which is directly related to the

Learning Objective.

8. Assessing the Performance

- In mathematics classes, assessing student

performance usually takes the form of quizzes and

tests. - With every such assessment tool the instructor

must be concerned with reliability and validity.

8. Assessing the Performance

- Is the observation reliable or was the correct

response obtained by chance? - Notice that reliability is low for true/false and

multiple choice questions unless special care is

taken in the construction of such questions.

8. Assessing the Performance

- Is the observation valid ?
- That is does correct performance accurately

reflect the objective?

8. Assessing the Performance

- Is the observation of correct performance free of

distortion? - Distortion might be memorization of an answer or

recall of an answer from some previous example. - Notice that when a test contains exercises which

the student has previously seen, the chance for

distortion is great.

9. Enhancing Retention and Transfer

- Arranging a variety of practice to aid future

retrieval and transfer. - If concepts, rules, procedures, etc. are to be

well retained, provision must be made for

systematic reviews spaced at intervals of weeks

and months. This is more effective than repeated

examples immediately following the initial

learning.

Enhancing Retention and TransferMathematics

- What this means in mathematics is that it is

probably ineffective to do large numbers of

exercises during or immediately after the initial

learning.

Enhancing Retention and TransferMathematics

- It is more effective to
- Regularly review those items (definitions, rules,

concepts, etc.) which are to be memorized and - Regularly go back to previous sections and work a

few exercises