Approaches, Methods, and Techniques

Fourth Topic

5-E Learning Cycle

an instructional model based on constructivist approach, emphasizing active student engagement and discovery. Consist of five phases, engage, explore, explain, elaborate, and evaluate. Allows student to build their understanding through hands-on experiences and structured inquiry, making mathematics more meaningful and applicable.

Engage

teacher introduces a problem, question, or activity that captures student curiosity at the same time connects prior knowledge with new concepts. eg. Presenting a real life scenario such as how can evenly divide pizza among different groups of friends?

Explore

students engage in hands-on activities to investigate the concept. They make observations, collect data, and attempt problem-solving strategies. The teacher facilitates but does not directly instruct. eg. using manipulatives like fraction strips or cut out circles to explore fractions physically.

Explain

students share their feelings and the teacher provides structured explanations. new vocabulary and mathematical concepts are formally introduced. eg. defining fractions, denominators and numerators after students have manipulated physical models.

Elaborate

students apply the learned concepts to new situations and deepen understanding. The teacher encourages problem-solving and higher-order thinking skills. eg. solving word problems involving fraction addition or designing their own fraction problems.

Evaluate

The teacher assesses the student’s understanding using various tools (quizzes, projects, self-assessments). Students reflect on what they learned. eg. asking students to explain how they would divide a cake among seven people using diagrams.

Activity Approach

a student-centered strategy that engages learners in hands-on interactive experiences to develop mathematical understanding. They explore, discover, and apply concepts through real world activities, experiments, problem-solving and collaborative tasks.

Using Patterns

refer to systematic approaches that help students recognize, analyze, and extend patterns. Involve observing relationships, identifying rules, and making predictions based on repeated structures in numbers, shapes, or real-world contexts. Help students develop algebraic thinking, problem solving skills, and logical reasoning.

Numerical, shape and spatial, algebraic, symmetry

types of patterns in mathematics

Inductive approach

an approach where students discover patterns through observation and exploration. eg. present a series of number patterns and let students find the rule

Deductive Approach

the teacher provides the rule first, and students apply it to examples eg. explain the rule for an arithmetic sequence before giving students problem.

Constructivist Approach

students build their own understanding by exploring real-world patterns. eg. investigating how patterns appear in nature such as in animals or things.

Problem-solving approach

students solve problems using patterns as a tool for reasoning. eg. finding the number of seats in a triangular arrangement using a pattern

Investigative Approach

teaching method that encourages students to explore mathematical concepts through inquiry, experimentation and discovery. instead of directly providing the formula or rules, teachers guide students to observe patterns, test ideas, and make conclusions based on their findings.

Pose questions, make conjectures, test and experiment, analyze and justify, apply and extend

these are encourage in the investigative approach.

Cooperative Learning

a student-centered approach where learners actively collaborate in small groups to understand mathematical concepts and solve problems. Fosters peer interactions, collective problem solving, and high levels of student engagement in the learning process.

Enhanced Understanding, Increased Confidence, Active Participation, Improved Skills

Importance of Cooperative Learning in Mathematics

Student Teams-Achievement divisions

students work together in teams to master specific concepts, and individual assessments are used to measure progress

Jigsaw Technique

each member of the group learns a different part of a topic and then teaches it to their peers, ensuring everyone understand all aspects of the subject.

think-pair share

students individually reflect on a problem, discuss it with a partner, and then share their solutions with the class, promoting engagement

Group investigation

groups research a particular topic, discuss their findings, and present their conclusions to the class, fostering deeper exploration

reciprocal teaching

students take turns being the teacher, leading discussions and explaining mathematical ideas to their peers.

peer tutoring

more knowledgeable students assist their peers in understanding complex concepts, reinforcing their own learning in the process.

Problem-based Learning

small groups collaborate to solve real world mathematical problems, encouraging application of skills. It requires students to apply strategies to tackle different types of math problems. Search polyas method.

collaborative worksheets

groups work together to complete mathematical problem sets, ensuring that every member contributes to the solution process.

Give specific goals for each group.

Assign specific responsibilities

Use well planned activities

Guide group interactions and correct any misunderstandings during the process

Provide feedback like constructive feedback (what worked well and what would you change next time?)

techniques for effective cooperative learning in mathematics

Project-based Learning

an approach where students learn by engaging in real-world projects that require them to apply their knowledge and skills eg. making a mini playground with geometrical shapes display.

Define clear learning goals (should be aligned with the curriculum)

use authentic, real world problems

encourage inquiry and exploration

provide scaffolding and support

integrate technology and hands-on activiities

techniques for effective pbl in math

Understand, Plan, Solve and Reflect

polya’s four-step process

Polyas Four step process (understand, plan, solve, reflect)

encourage visualization (graphs, diagrams, models)

use hands-on learning (real world examples)

teach multiple solution strategies (promote trial and error method)

promote metacognitive reflection (analyze their thinking)

techniques for effective problem solving

> break down the problem (make it manageable to digest)

> draw pictures (making complex concepts more concrete and understandable)

methods and technique for problem solving

model ps techniques

asking guide questions

encouragement

role of teachers in teaching ps

Discovery Approach

active hands-on style of learning discovered by Jerome Bruner in 1960s

Inquiry Approach

a primary pedagogical method, developed during the discovery learning movement of the 1960 as a response to traditional forms of instructions. Key contributors include John Dewey

Process Approach

emphasizes the importance of understanding the processes involved in learning and problem-solving, rather than solely focusing on the final product or correct answer. Key contributors: Lev Vygotsky.

Activity-Based Method

learning happens through hands-on activities and experiments. eg. using number tiles to form multiplication arrays instead of memorizing tables.

Socratic Questioning

technique used in the inquiry method approach where the facilitator ask deep questions to encourage reasoning and for students to arrive on the answer. eg. why does this pattern exist? (this promotes deeper answers enabling connections to prior knowledge and reasons)

Concrete Representational Abstract approach (kumbaga explore before explain)

used to effectively integrate discovery based math. includes beginning with concrete materials like real-life objects or models to explore concepts. eg. using base 10 blocks to represent values, then transition to representational stage where students use pictures, images or virtual manipulatives to represent concrete materials.

Incorporate interactive content using technology

used to effectively integrate inquiry approach in classroom. Use of simulations and quizzes online to discover concepts through models.