Unit-VI

Fennema and Franke (1992) determined the components of mathematics teachers’ knowledge as;

1) Knowledge of mathematics, Content knowledge, The nature of mathematics, & The mental organization of teacher knowledge

2) Knowledge of mathematical representations

3) Knowledge of students & Knowledge of students’ cognitions

4) Knowledge of teaching and decision-making.

Knowledge of mathematics and knowledge of mathematical representations are related to content knowledge, while knowledge of students and knowledge of teaching are related to pedagogical content knowledge.

According to An, Kulm, and Wu (2004) pedagogical content knowledge has three components:

• Knowledge of content

• Knowledge of the curriculum

• Knowledge of teaching

However, they also outlined that the content of pedagogical content knowledge is ‘content-specific and at the same time goes beyond simple knowledge of mathematics therefore a mathematician may not posses it (Kahan, Cooper and Bethea, 2003: 223)’.

A number of factors may influence the teaching of mathematics but teachers play an important role in the teaching process. The common belief in society is if a mathematics teacher knows mathematics very well, he or she is the best person to teach mathematics. But what about “knowing to teach mathematics”?

Pedagogical analysis of mathematics involves breaking down the content of mathematics into teachable units, considering the methods and strategies that can be used to teach these units effectively, and understanding the difficulties students might face while learning these concepts. Here's an in-depth look at the components of a pedagogical analysis of mathematics:

1.Content Analysis:

Content analysis in the context of mathematics education involves breaking down the curriculum into manageable and teachable units. This analysis helps teachers understand the structure of the subject matter, identify key concepts, and plan effective lessons. Here's a detailed framework for conducting content analysis in mathematics:

·   Dentification of Key Concepts: Determine the main topics and subtopics within the mathematics curriculum. For example, algebra, geometry, calculus, etc.

·   Hierarchy of Concepts: Establish the prerequisite knowledge required for understanding more advanced topics. For instance, understanding basic arithmetic before moving on to algebra.

·   Conceptual Mapping: Create a visual representation of how different mathematical concepts are related.

2. Objective Setting:

Instructional objectives are specific, measurable statements that clearly define what students will know and be able to do as a result of instruction. They provide a clear direction for both teaching and learning and help to assess student progress effectively. Here's how to craft and use instructional objectives in the context of a mathematics course:

·   Cognitive Objectives: Define what students should know and understand. For example, students should be able to solve linear equations.

·   Affective Objectives: Define what students should feel or appreciate. For instance, students should appreciate the beauty of mathematical patterns.

·   Psychomotor Objectives: Define what students should be able to do. For example, students should be able to construct geometric shapes using tools.

3. Teaching Methods and Strategies:

Teaching methods and strategies refer to the various approaches and techniques educators use to facilitate learning and help students achieve their instructional objectives. Effective teaching methods are essential for engaging students, promoting understanding, and fostering critical thinking skills. Here are some commonly used teaching methods and strategies in education:

·   Direct Instruction: Use of Inductive, demonstrations, and step-by-step problem-solving.

·   Inquiry-Based Learning: Encourage students to ask questions and discover solutions through exploration.

·   Collaborative Learning: Promote group work and peer-to-peer teaching.

·   Use of Technology: Incorporate tools such as graphing calculators, educational software, and online resources.

·   Manipulatives and Visual Aids: Use physical objects and visual representations to help students understand abstract concepts.

4. Assessment and Evaluation:

Assessment and evaluation are crucial components of the educational process, providing insights into student learning, instructional effectiveness, and curriculum alignment. They serve to measure and improve student performance, guide instructional decisions, and ensure educational standards are met.

·   Formative Assessment: Use quizzes, class activities, and homework to monitor student progress.

·   Summative Assessment: Implement tests, exams, and projects to evaluate overall understanding.

·   Feedback Mechanisms: Provide timely and constructive feedback to help students improve.

·   Self-Assessment: Encourage students to reflect on their learning and identify areas for improvement.

By conducting a pedagogical analysis of mathematics, educators can create comprehensive lesson plans that address the needs of all students, making mathematics more accessible and enjoyable to learn.

Instructional objectives (also known as behavioral objectives or learning objectives) are basically statements that clearly describe an anticipated learning outcome. When objectives were first coming into their own in education, they almost always began with the phrase: "Upon completion of this lesson, the student should be able to…." This phrase focused on the outcome of learning rather than on the learning process. In fact, one of the criteria for a well-written objective is that it describes the outcome of learning, that is, what the learners can do after learning has occurred that they might not have been able to do before the teaching and learning process began.

Characteristics of a Well-Written Objective

A well-written objective should meet the following criteria:

(1) describe a learning outcome,

(2) be student oriented,

(3) be observable (or describe an observable product).

A well-written objective should describe a learning outcome (e.g., to correctly spell the spelling of the words on page-17 of English text book). It should not describe a learning activity (e.g., to practice the words on page-17 by writing each one ten times). Learning activities are important in planning and guiding instruction but they are not to be confused with instructional objectives.

Characteristics of Instructional Objective

To be useful for instruction, an objective must not only be well written but it also must meet the following criteria:

(1) be sequentially appropriate;

(2) be attainable within a reasonable amount of time;

(3) be developmentally appropriate.

Kinds of Instructional Objectives

Instructional objectives are often classified according to the kind or level of learning that is required in order to reach them. There are numerous taxonomies of instructional objectives; the most common taxonomy was developed by Benjamin Bloom and his colleagues. The first level of the taxonomy divides objectives into three categories: cognitive, affective, and psychomotor. Simply put, cognitive objectives focus on the mind; affective objectives focus on emotions or affect; and psychomotor objectives focus on the body.

Cognitive objectives call for outcomes of mental activity such as memorizing, reading, problem solving, analyzing, synthesizing, and drawing conclusions. Bloom and others further categorize cognitive objectives into various levels from the simplest cognitive tasks to the most complex cognitive task. These categories can be helpful when trying to order objectives so they are sequentially appropriate. This helps to insure that prerequisite outcomes are accomplished first.

Affective objectives focus on emotions. Whenever a person seeks to learn to react in an appropriate way emotionally, there is some thinking going on. What distinguishes affective objectives from cognitive objectives is the fact that the goal of affective objectives is some kind of affective behavior or the product of an affect (e.g., an attitude). The goal of cognitive objectives, on the other hand, is some kind of cognitive response or the product of a cognitive response (e.g., a problem solved).

Psychomotor objectives focus on the body and the goal of these objectives is the control or manipulation of the muscular skeletal system or some part of it (e.g., dancing, writing, tumbling, passing a ball, and drawing). All skills requiring fine or gross motor coordination fall into the psychomotor category. To learn a motor skill requires some cognition. However, the ultimate goal is not the cognitive aspects of the skill such as memorizing the steps to take. The ultimate goal is the control of muscles or muscle groups.

General Instructional Objective: An intended outcome of instruction that has been stated in general enough terms to encompass a domain of student performance 

Specific Learning Objectives: A goal or aim serves as a guide for a teaching unit, directed toward the eventual achievement of a general objective. It is stated in terms of specific and observable pupil performance. It describes the specific behaviour, a learner will exhibit after learn­ing a particular unit. Let us discuss separately different criteria for stating, general instructional objectives and specific learning objectives.

Examples:

General Objective: Understand graphs and charts.

Specific Learning Objective: Distinguish between graph and chart.

THE ROLE OF ROBERT MAGER IN WRITING INSTRUCTIONAL OBJECTIVES

Robert Mager (1997) is a renowned educational psychologist who made significant contributions to the field of instructional design. His work on writing clear and measurable instructional objectives has been widely adopted and has influenced educational practices globally. Mager's approach emphasizes precision, clarity, and measurability, ensuring that both instructors and learners have a clear understanding of the desired learning outcomes.

Key Contributions of Robert Mager

1. Clarification of Objectives:

   • Mager stressed the importance of writing instructional objectives that are clear and unambiguous. This clarity helps ensure that both teachers and students understand what is expected.

2. Focus on Measurable Outcomes:

   • Mager advocated for the use of specific, observable, and measurable terms in instructional objectives. This approach helps in assessing whether the objectives have been achieved.

3. Three-Part Structure:

   • Mager introduced a three-part structure for instructional objectives, which includes:

     • Performance (Behavior): Describes what the learner will be able to do.

     • Conditions: Specifies the conditions under which the performance will occur.

     • Criterion: Defines the standard of performance that is considered acceptable.

Mager's Three-Part Structure Explained

1. Performance (Behavior):

   • Description: This component specifies the actual task or behavior the learner is expected to perform. It should be an action that can be observed and measured.

   • Example: "The student will be able to solve quadratic equations."

2. Conditions:

   • Description: This component describes the circumstances under which the learner will perform the task. It includes any tools, resources, or constraints that are part of the task.

   • Example: "Given a set of quadratic equations."

3. Criterion:

   • Description: This component specifies the level of performance that will be considered acceptable. It defines how well the learner must perform the task to meet the objective.

   • Example: "With 90% accuracy."

Steps to Writing Instructional Objectives Using Mager's Approach

1. Identify Desired Outcomes:

   • Determine what you want learners to achieve by the end of the instruction. Be specific about the skills, knowledge, or attitudes you expect them to develop.

2. Use Action Verbs:

   • Choose verbs that describe observable and measurable actions. Avoid vague terms like "understand" or "know." Instead, use verbs like "analyze," "describe," "solve," or "create."

3. Specify Conditions:

   • Clearly state the conditions under which the learner will perform the task. This could include the tools they will use, the environment, or any other relevant factors.

4. Set Performance Criteria:

   • Define the criteria for acceptable performance. Specify how well the learner must perform the task to meet the objective. This could include accuracy, speed, quality, or other relevant measures.

5. Write the Objective:

   • Combine the performance, conditions, and criteria into a single, clear statement.

   • Example: "Given a set of quadratic equations, the student will be able to solve them with 90% accuracy."

Example of an Instructional Objective

1. Mathematics:

   • Objective: "Given a worksheet of linear equations (Condition), the student will be able to solve them (Performance) with 100% accuracy (Criterion)."

2. Science:

   • Objective: "Using a microscope and prepared slides (Condition), the student will be able to identify parts of a plant cell (Performance) with at least 90% accuracy (Criterion)."

ROBERT MILLER APPROACH FOR WRITING INSTRUCTIONAL OBJECTIVES

Robert Miller (1962) concept of instructional objectives are based on research on training process in military and related to development of skills and physical activities.

He gave action verbs only for psychomotor domain.(based on Harrow's classification).  There are 3 essential elements acc.to him:

·   Indication(stimulus)

·   Activation (response to be made)

·   Feedback(info of made response)

Dr. Robert Miller (1962) put forward his scheme based on skill analysis:

1. Description of the indicator, indicating the relevant activity

2. Description of the indication or stimulus that calls for a responses

3. Controlling of the object that is to be activated

4. Description of the activity to be performed

5. The indication of the adequacy of responses or feedback.

NCERT APPROACH FOR WRITING INSTRUCTIONAL OBJECTIVES

The NCERT (National Council of Educational Research and Training) approach to writing instructional objectives focuses on creating clear, concise, and comprehensive goals for student learning. This approach ensures that the educational objectives are aligned with the curriculum and help in achieving the desired learning outcomes. Here are the key aspects of the NCERT approach for writing instructional objectives:

1. Clarity and Specificity:

• Characteristics: Objectives should be clear and specific, detailing what the students are expected to learn and achieve. They should be free of ambiguity and easily understandable.

• Example: "Students will identify the different parts of a plant."

2. Measurability:

• Characteristics: Objectives should be measurable, meaning there should be a clear way to assess whether the students have achieved the objective.

• Example: "Students will solve 10 out of 12 problems related to fractions accurately."

3. Achievability:

• Characteristics: Objectives should be realistic and attainable within the given time frame and resources. They should consider the students' prior knowledge and abilities.

• Example: "Students will write a short essay on their favorite animal, including at least three specific details."

4. Relevance and Alignment:

• Characteristics: Objectives should be relevant to the students' needs and interests and aligned with the overall curriculum goals and standards.

• Example: "Students will understand the importance of recycling and be able to list five recyclable materials."

5. Time-bound:

• Characteristics: Objectives should specify the time frame within which the students are expected to achieve the learning goals.

• Example: "By the end of the week, students will perform basic addition and subtraction operations."

NCERT's Specific Framework for Instructional Objectives:

1. Knowledge Objectives:

   • Characteristics: Focus on recalling facts, terms, basic concepts, and answers.

   • Example: "Students will recall the major rivers of India."

2. Understanding Objectives:

   • Characteristics: Focus on understanding and interpreting meanings, translations, and instructions.

   • Example: "Students will explain the water cycle process."

3. Application Objectives:

   • Characteristics: Focus on using information, concepts, and principles in new situations.

   • Example: "Students will solve problems on area in real-life involving rectangular plots."

4. Analysis Objectives:

   • Characteristics: Focus on breaking down information into parts to understand its structure.

   • Example: "Students will analyze the causes and effects of pollution."

5. Synthesis Objectives:

   • Characteristics: Focus on compiling information in a different way by combining elements in a new pattern.

   • Example: "Students will create a poster promoting environmental conservation."

6. Evaluation Objectives:

   • Characteristics: Focus on making judgments about the value of ideas or materials.

   • Example: "Students will judge the effectiveness of different recycling methods."

By following these principles, the NCERT approach ensures that instructional objectives are well-defined, achievable, and aligned with educational standards, ultimately contributing to effective teaching and meaningful learning.

Teaching aids provide a stimulus for exploration end thinking. With the added input of verbal, personal communication with an adult, interaction and discussion arise...and these are crucial to reel, activity-based learning. Adults (and older children) help younger ones to interpret sensory and language experiences, to clarify them and relate them to their previous understandings. Children then learn by blending language with experiences. They learn to think.

For maximum mental growth and personality development, a child’s life needs to be filled with stimulating, encouraging experiences. Appropriate learning materials (teaching aids) help children to develop their innate abilities.

Concept of Teaching Aids:

Teaching aids are known as audio-visual aids, and sensory aids. Helps to make teaching concrete, effective, and interesting. Cannot replace a teacher. Communication of ideas between persons & groups.

Sources of Learning:     Direct sensory contact, Pictures or similar other representation of facts, Oral or printed words

Types of Teaching Aids: Following are some of the classifications 

Classification – 1:

a) Audio Aids: Audio aids mean learning through listening. Audio tape, Radio, Tape Recorder etc.    

b) Visual Aids: Visual aids mean Learning through seeing. LCD projector, Chart, Model, Picture, Chalk Black Board,  etc.

c) Audio–visual aids are devices that can be heard or as well as seen. This aids to impart knowledge through senses to ensure quick and effective learning. Video with Audio, TV etc.

    Classification – 2:

1. Projected Aids: Aids which help in their projection on the screen are called projective aids. When a projected aid is used and enlarged image of the material (slide, filmstrips etc) is projected on a screen kept at a distance from the projector. 
2. Non Projected Aids : Visual instructional devices which are simply presented without any projection equipment are non projected aids. Examples: blackboard, chart, etc. Projected Aids: visual instructional devices which are shown with a projector are called projected aids.