MATHEMATICS PDE 115

UNIT ONE: ELEMENTS OF PHILOSOPHY OF MATHEMATICS AND PROOF IN MATHEMATICS

1. Introduction

Point: Philosophy of mathematics studies the nature, role, and methodology of mathematics.
Explanation: It asks questions like: “Why is mathematics useful in describing nature?” or “How do mathematical statements exist?” This helps us understand mathematics deeply and affects how it is taught and learned. Unlike everyday life, mathematics requires rigorous proof, not just observation.

Question: Why is philosophy of mathematics important?
Answer: It helps define the nature, role, and teaching methodology of mathematics.

2. Objectives of the Unit

By the end of this unit, you should be able to:

1. Distinguish between philosophy of mathematics and mathematics education.
2. Differentiate logicism, formalism, and intuitionism.
3. Discuss the influence of philosophy on mathematics teaching.
4. Explain what a proof is.
5. Give reasons why proofs are necessary.
6. Describe some proof methods.

Question: Name three objectives of this unit.
Answer: Distinguish philosophy of mathematics from math education, explain what a proof is, and describe proof methods.

3. Philosophy of Mathematics vs. Mathematics Education

Point: Philosophy of mathematics studies the nature of mathematics; philosophy of math education studies how beliefs about math influence teaching and learning.
Explanation: Individuals and societies hold beliefs about mathematics that affect classroom instruction. For example, some may see math as fixed knowledge (Platonism) while others see it as constructed by humans (quasi-empirical).

Question: How does philosophy of mathematics influence teaching?
Answer: It shapes how teachers approach instruction, whether as fixed knowledge or as knowledge to be constructed by learners.

4. Foundationalist Schools of Mathematics

Point: The three major foundationalist schools are Logicism, Formalism, and Intuitionism.

Explanation:

• Logicism: Math is a set of abstract truths derived from logic. Criticism: too rigid and limits creative thinking.
• Formalism: Math is about manipulating symbols according to rules. Criticism: little room for intuition or meaning.
• Intuitionism: Math concepts are mental constructions guided by natural laws. Criticism: too individualistic and can ignore social aspects of math.

Question: Name the three foundationalist schools of mathematics.
Answer: Logicism, Formalism, Intuitionism.

5. Quasi-Empirical Mathematics

Point: Mathematics can be viewed as fallible, socially constructed, and evolving.
Explanation: Unlike pure mathematics, quasi-empirical math emphasizes application, experimentation, and revision. Truths in mathematics may depend on time and context.

Question: What is quasi-empirical mathematics?
Answer: A view that mathematics is human-created, fallible, and dependent on practical and social context.

6. Influence on Mathematics Pedagogy

Point: Philosophy of mathematics shapes teaching methods.
Explanation:

• Foundationalist ideas led to the “New Math” emphasizing symbols, logic, and analytical thinking.
• Quasi-empirical views encourage problem-solving, exploration, and learner-centered teaching.

Question: How does the quasi-empirical view influence teaching?
Answer: It encourages active learning, problem-solving, and constructing knowledge rather than just memorizing facts.

7. Constructivism vs Behaviorism

Point: Two major teaching philosophies: constructivism and behaviorism.
Explanation:

• Constructivism: Learners actively construct knowledge using prior experiences. Focus on understanding and higher-order thinking.
• Behaviorism: Learning is shaped by rewards, repetition, and mastering skills. Focus on rote learning and correct answers.

Question: Give one difference between constructivism and behaviorism.
Answer: Constructivism emphasizes learner understanding, while behaviorism emphasizes rote memorization and repetition.

8. What is a Proof

Point: A proof is a logical argument that shows a mathematical statement is true.
Explanation: Proofs are based on definitions, premises, postulates, and previously established results. They can include diagrams, words, symbols, or computer programs.

Question: What is a proof in mathematics?
Answer: A logical argument that establishes the truth of a mathematical statement.

9. Why Do We Prove?

Point: Proofs are done to:

1. Establish truth with certainty.
2. Gain understanding of a problem.
3. Communicate ideas to others.
4. Enjoy the challenge of problem-solving.
5. Create something beautiful in mathematics.
6. Develop mathematical language and theory.

Question: Give two reasons why mathematicians prove statements.
Answer: To establish truth with certainty and to communicate ideas to others.

10. Proof Methods

Point: Common proof methods include:

• Proof by induction
• Proof by contradiction
• Pigeonhole principle
• Parity argument
• Invariants
• Example as disproof
• Proof by exhaustion
• Proof pending a lemma

Question: Name three proof methods.
Answer: Proof by induction, proof by contradiction, proof by exhaustion.

11. Lemma and Corollary

Point:

• Lemma: A minor result used to help prove a larger theorem.
• Corollary: A result that follows easily from a previously proven theorem.

Question: Distinguish lemma and corollary.
Answer: A lemma is a helper result; a corollary follows directly from a theorem.

12. Ending a Proof

Point: Proofs are traditionally ended with “Q.E.D.” (quod erat demonstrandum) or “Q.E.A.” (quod est absurdum). Modern styles may use symbols or informal endings.

Question: How do mathematicians end proofs?
Answer: Traditionally with “Q.E.D.” or “Q.E.A.” or a symbol to indicate completion.

13. Example Proof

Point: Sum of angles on a straight line equals two right angles.

Proof:
Let a straight line AB have a point C. Angles on the line are ∠ACB and ∠BC. By definition of straight line, ∠ACB + ∠BC = 180° = 2 right angles.

Question: Prove that the sum of angles on a straight line is 180°.
Answer: See proof above.

Activity Questions

1. What are the foundationalist schools of thought in mathematics education?
2. Distinguish between constructivist and behaviorist views with examples.
3. Why is proof important in mathematics?
4. Give at least seven proof methods.
5. How do we end a proof?
6. Prove that the sum of angles on a straight line equals two right angles.

UNIT 3: COGNITIVE THEORIES OF LEARNING AND MATHEMATICS TEACHING

Introduction

Mathematics teaching is guided by different learning theories. These theories explain how students learn, help in designing lessons, and address learning difficulties. This unit focuses on cognitive theories, teaching methods, and common problems in math education.

Objectives

By the end of this unit, you should be able to:

1.      Explain the cognitive theory of learning.

2.      Describe the theories of Gagne, Ausubel, Piaget, and Bruner in relation to math teaching.

3.      Identify methods for teaching mathematics.

4.      Discuss common challenges in teaching mathematics.

5.      Mention at least 8 methods of teaching mathematics.

6.      Describe at least 6 methods of teaching mathematics.

7.      Write lesson notes for topics using 6 methods.

How to Study

·         Read carefully and relate concepts to past classroom experiences.

·         Practice teaching skills in micro-teaching exercises.

·         Attempt all activities.

Cognitive Theories of Learning

Cognitive theories focus on mental processes: how learners process, store, and recall information. Students actively construct knowledge by linking new information to what they already know.

Key Theorists

1.      Jerome Bruner – Discovery Learning

o    Teach the structure of a subject.

o    Three stages of learning:

1.      Enactive – learning through actions or objects.

2.      Iconic – learning with models and pictures.

3.      Symbolic – abstract thinking.

o    Principle: Concrete → Pictorial → Symbolic leads to effective learning.

o    Emphasizes discovery learning, engaging students actively in finding solutions.

2.      David Ausubel – Advance Organizer

o    Focuses on meaningful reception learning.

o    Advance organizer: helps students connect new knowledge to existing knowledge.

o    Encourages active learning: underlining, rewording, or providing examples.

3.      Robert Gagne – Events of Learning

o    Views learning as a sequence of 9 events:

1.      Gain attention

2.      Inform objectives

3.      Recall prior learning

4.      Present new material

5.      Guide learning

6.      Elicit performance

7.      Provide feedback

8.      Assess performance

9.      Enhance retention and recall

o    Emphasizes learning hierarchies: lower-level knowledge must precede higher-level concepts.

o    Classifies learning into 5 categories: verbal info, intellectual skills, cognitive strategies, motor skills, attitudes.

Methods of Teaching Mathematics

Methods focus on how content is structured and how it is presented. They are not always mutually exclusive.

1. Synthetic vs Analytic

·         Synthetic: From known → unknown (combining truths).

·         Analytic: From unknown → known (breaking into simpler truths).

·         Example: Solving quadratic equations step by step analytically.

2. Deductive vs Inductive

·         Deductive: General → Particular (e.g., All men are mortal → Socrates is mortal).

·         Inductive: Particular → General (e.g., Observing patterns to form a rule).

3. Discovery Method

·         Student-centered; teacher guides, but students explore and find solutions.

·         Advantages: Long-lasting learning, intrinsic satisfaction, encourages inquiry.

·         Disadvantages: Time-consuming, less coverage, may discourage below-average students.

4. Heuristic Method

·         Students individually discover facts without teacher guidance.

·         Advantages: Active participation, develops problem-solving skills.

·         Disadvantages: Time-consuming, may be challenging for low-IQ students.

5. Analytic Method

·         Requires prior understanding and knowledge.

·         Student analyzes problems and creates solutions (algorithm).

6. Synthetic Method

·         Combines known information to solve problems.

·         Suitable for higher secondary or tertiary levels.

7. Laboratory Method

·         Students perform experiments, play math games, investigate patterns.

·         Develops inquiry skills and understanding through hands-on learning.

·         Stages:

1.      Free exploration

2.      Directed investigation

3.      Practice session

Learning Aids in Mathematics

·         Manipulatives: Real-world objects (counters, blocks, coins) to show math concepts.

·         Help students connect concrete experience with symbolic math.

·         Improper use can confuse students between real-world and symbolic math.

Activity

1.      Choose a math topic from primary to SS3 syllabus.

2.      Apply each teaching method (synthetic, analytic, discovery, heuristic, laboratory, etc.) to show how it can be taught effectively.

Here’s a simplified counseling/summary of your additional Unit 3 content on manipulative materials and learning problems in mathematics, ready for easy reading and Opera Mini posting:

Manipulative Materials in Mathematics

Definition

• Manipulative materials are concrete objects that students can touch, feel, and move to understand mathematical concepts.
• They are not just teacher demonstrations but tools for active student learning.
• Examples suitable for Nigerian classrooms: stones, beans, tins, oranges, mangoes, matchsticks.
• Not suitable: abacus (not commonly used in daily life).

Principles for Using Manipulatives

• Each student should manipulate materials independently.
• Materials should match the concept and developmental level of students.
• Example: Use 10 sticks bundled together for the tens place; 100 sticks for hundreds.
• Good manipulatives are durable, simple, attractive, and manageable.
• Storage: baskets or boxes for easy access.

Benefits for Students

Using manipulatives helps students:

1. Connect real-world situations to math symbols.
2. Work cooperatively.
3. Discuss math concepts.
4. Verbalize their thinking.
5. Present solutions in front of a group.
6. Solve problems in multiple ways.
7. Represent problems symbolically in different forms.
8. Gain independence from teacher instructions.

Assessment with Manipulatives

• Focus less on testing; more on concept development and understanding.
• Evaluate students by:
1. Listening to their mathematical thinking.
2. Observing individual and group work.
3. Asking why/how questions instead of just yes/no or numeric answers.
4. Having students write solutions to show understanding.

Examples of Assessment Questions:

• How do you know that?
• What would happen if…?
• Why do you think 2…?
• How could you prove your answer?
• Can you solve it in a different way?
• How can you convince others your method is best?

Why Some Students Struggle in Mathematics

Learning Characteristics

Students with learning problems may experience:

1. Learned Helplessness – believing they cannot learn due to repeated failure.
2. Passive Learning – unwilling to take risks or apply knowledge.
3. Memory Problems – difficulty recalling steps in problem-solving.
4. Attention Problems – missing critical steps in procedures.
5. Metacognitive Deficits – inability to evaluate and adjust strategies.

Instructional Issues

• Spiral Curriculum: Students move to new concepts before mastering old ones.
• Algorithm-only vs Concept-only Teaching: Teaching only algorithms or only concepts leaves gaps.
• Cyclical Reforms: Frequent changes in teaching methods confuse students with learning problems.

Effective Teaching Practices

• Use research-based strategies that match students’ unique learning needs.
• Combine concept understanding with algorithm skills.
• Systematically implement strategies in textbooks and classroom instruction.

Activity

1. What are manipulative materials?
2. Why are manipulatives necessary in math teaching?
3. Enumerate barriers to mathematics success.
4. Discuss instructional issues that affect students with learning problems.

Summary

• Student enthusiasm and engagement are key to learning.
• Teachers can explain, consult, or give extra lessons, but learning is effective only when students actively try to understand.
• Manipulatives, discovery, and active learning methods enhance understanding and conceptual development in mathematics.

 

 

UNIT 4: Educational Objectives, Lesson Planning, Continuous Assessment & Test Construction

Introduction

• Teaching or testing without clear objectives is like starting a journey without knowing the destination.
• Proper planning ensures effective teaching and learning.

Objectives of This Unit

By the end of this unit, you should be able to:

1. Distinguish between general and specific objectives.
2. Define behavioral objectives.
3. Explain Bloom’s Taxonomy of Educational Objectives.
4. Define a Table of Specification (TOS).
5. Draw a TOS for a topic or multiple topics.
6. State the advantages of a TOS.

General vs Specific Objectives

• General Objectives: Broad aims of education (e.g., why we study mathematics).
• Specific Objectives: Focused on one skill, knowledge, or attitude.

Behavioral Objectives

• Should be observable and measurable.
• Use active verbs like add, solve, draw, state.
• Avoid passive verbs like know or understand.

Examples:

• Solve quadratic equations using completing the square.
• Draw a graph of a quadratic equation.
• State the Pythagoras theorem.

Bloom’s Taxonomy of Educational Objectives

1. Cognitive Domain – Mental skills: knowledge → comprehension → application → analysis → synthesis → evaluation.
2. Affective Domain – Emotions, attitudes, values.
3. Psychomotor Domain – Physical skills: writing, typing, manipulating objects.

Levels of Cognitive Domain:

• Knowledge: Recall facts (e.g., sum of angles of a triangle).
• Comprehension: Understand meaning (e.g., explain a rule).
• Application: Use knowledge in new situations.
• Analysis: Break information into parts and study relationships.
• Synthesis: Create new ideas or procedures.
• Evaluation: Judge or assess based on criteria.

Lesson Planning

Two kinds of lesson plans:

1. Unit Plan: Covers a topic over several days/weeks.
• Includes: topic, objectives, teaching materials, pre-test, content, method, assessment, references.
2. Daily Plan: Focuses on one day’s lesson, based on the unit plan.
• Includes: objectives, teaching materials, basic knowledge, content, method, assessment, teacher’s feedback.

Example: Everyday Statistics (Primary IV)

• Unit Plan: 3 lessons – vertical pictogram, horizontal pictogram, identifying most common object.
• Daily Plan: Vertical pictogram – pupils divide 10 matchboxes among 3 students, arrange vertically, and read the pictogram.

Assessment: Pupils read pictogram on board; Assignment: bring 5 pebbles next day.

Continuous Assessment (CA)

• Definition: Systematic evaluation using varied tools to monitor learner performance continuously.
• Characteristics: Systematic, comprehensive, cumulative, guidance-oriented, well-monitored.
• Advantages:
• Helps teacher participate in student learning.
• Encourages better preparation for both teacher and student.
• Reduces exam malpractice.
• Assesses cognitive, affective, psychomotor domains.

Phases of CA:

1. Entry Phase: Identify interests, background, abilities.
2. Passage Phase: Monitor development, progress, learning experiences.
3. Terminal Phase: Evaluate aptitude, skills, and readiness for next level.

Record Keeping: Use Progress Record Card – personal info, academic report, psychomotor and affective performance, terminal exams, yearly summary.

Problems with CA:

• Difficulty assessing non-cognitive domains.
• Misconception that CA is just testing cognitive skills.
• Lack of teacher training in CA techniques.

Recommendations:

• Conduct teacher workshops.
• Provide assessment tools from universities.
• Publish newsletters and periodicals on CA.

Test Construction

• Tests should align with objectives taught.
• Table of Specification (TOS): Grid connecting content (rows) with behavioral objectives (columns).
• Ensures questions cover all topics and cognitive levels proportionally.

Advantages of TOS:

1. Ensures content validity – test measures what was taught.
2. Gives clear perspective of teaching objectives.
3. Provides diagnostic information for teachers and students.

Activities

1. Write behavioral objectives for:
• Recognition of numbers (Nursery I)
• Addition of two-digit numbers
• Ratio, Variation, Fractions, Decimals, Percentages
2. Categorize WAEC/NECO past questions under Bloom’s levels.
3. Write a lesson note to teach Simultaneous Equation to JSS 3.
4. List characteristics, advantages, and problems of continuous assessment.