Guiding Principles/Theories and Teaching Approaches and Methods in the Teaching of Math Educational Theories n which Math Teaching is Grounded Framework of Learning Theories o Experiential learning - is the process of learning through experience, and is more specifically defined as "learning through reflection on doing". o Constructivism - is a theory of knowledge[1] that argues that humans generate knowledge and meaning from an interaction between their experiences and their ideas. o Cooperative learning - aims to organize classroom activities into academic and social learning experiences. o Discovery – is a theory that takes place in problem solving situations where the learner draws on his or her own past experience and existing knowledge to discover facts and relationships and new truths to be learned. o Inquiry-based learning - Instead of just presenting the facts, use questions, problems, and scenarios, help students learn through their own agency and investigation. The Spiral Progression in the Teaching of Math - Basic principles are introduced in the first grade and are rediscovered in succeeding grades in more complex forms - Students continually return to basic ideas as new subjects and concepts are added over the course of a curriculum

Grades 7-10 Grades 4-6 Grades 1-3 K– Algebra, Geometry, Statistics And Probability

Principles of Teaching Math 1. Balance Principle Balance Approach a) Standard-based - i.e., concise, written descriptions of what students are expected to know and be able to do at a specific stage of their education—determine the goals of a lesson or course, and teachers then determine how and what to teach students so they achieve the learning expectations described in the standards. b) Integrated - The goal is to help students remain engaged and draw from multiple sets of skills, experiences and sources to aid and accelerate the learning process.

c) Engaging – (meaning charming, interesting, pleasing, appealing, attractive, lovely, fasc inating, entertaining, winning,pleasant d) Open to practices differentiated instruction - Differentiating instruction may mean teaching the same material to all students using a variety of instructional strategies, or it may require the teacher to deliver lessons at varying levels of difficulty based on the ability of each student. e) Makes use of problem-solving - Problem-solving is a tool, a skill, and a process. As a tool is helps you solve a problem or achieve a goal. As a skill you can use it repeatedly throughout your life. And, as a process it involves a number of steps. f) Guided practice - Guided Practice is interactive instruction between teacher and students. After the teacher introduces new learning, he/she begins the student practice process by engaging students in a similar task to what they will complete later in the lesson independently. Students and teacher collaboratively complete the task as a model. The teacher leads the activity but solicits help from students at predetermined points along the way. g) Makes use of manipulative, games and calculators - are physical tools of teaching, engaging students visually and physically with objects such as coins, blocks, puzzles, markers h) Assessment-driven and data driven instruction – assessment-data(determine students needs, A Practical Guide to Improve Instruction.) data-driven instruction (includes assessment, analysis, and action) 2. Three-tiered Principle Theory Definition Jerome Bruner theorized that learning occurs by going through three stages of representation. Each stage is a "way in which information or knowledge are stored and encoded in memory" (Mcleod, 2008). The stages are more-or-less sequential, although they are not necessarily age-related like Piaget-based theories. Going through the stages is essential to truly understanding the concept, as it helps the learner understand why. Bruner's Stages of Representation 1. enactive (action-based) Sometimes called the concrete stage, this first stage involves a tangible hands-on method of learning. Bruner believed that "learning begins with an action - touching, feeling, and manipulating" (Brahier, 2009, p. 52). In mathematics education, manipulatives are the concrete objects with which the actions are performed. Common

examples of manipulatives used in this stage in math education are algebra tiles, paper, coins, etc. - anything tangible. 2. iconic (image-based) Sometimes called the pictoral stage, this second stage involves images or other visuals to represent the concrete situation enacted in the first stage. One way of doing this is to simply draw images of the objects on paper or to picture them in one's head. Other ways could be through the use of shapes, diagrams, and graphs. video: iconic representation. Retreived fromhttp://www.youtube.com/watch?v=gIW0mjMo9IE 3. symbolic (language-based)Sometimes called the abstract stage, the last stage takes the images from the second stage and represents them using words and symbols. The use of words and symbols "allows a student to organize information in the mind by relating concepts together" (Brahier, 2009, p. 53). The words and symbols are abstractions, they do not necessarily have a direct connection to the information. For example, a number is a symbol used to describe how many of something there are, but the number in itself has little meaning without the understanding of it means for there to be that number of something. Other examples would be variables such as x or y, or mathematical symbols such as +, -, /, etc. Finally, language and words are another way to abstractly represent the idea. In the context of math, this could be the use of words such as addition, infinite, the number three, etc 3. For effective Math teaching it employ Experiential learning Constructivism Cooperative learning Discovery Inquiry-based learning Teaching Methods 1. Problem-Solving - consists of using generic or ad hoc methods, in an orderly manner, for finding solutions to problems. If there is a problem, there is a solution. Steps of the problem solving process a) Understanding the problem b) Planning and communicating a solution Other Techniques in Problem Solving a) Obtain the answer by trial and error b) Use an aid, model or sketch c) Search for a pattern

d) Elimination 2. Concept attainment strategy - engages students in forming their own definition of a concept by examining the attributes of several examples and non-examples of the word, concept, or topic. (ex. Giving a lot of examples and let them decide or think.) Separating important from unimportant information Searching for patterns and making generalization Defining and explaining concepts 3. Concept formation strategy - is, “a strategy that takes your students through a process whereby they work to understand a concept. Rather than you telling them, the students form their understanding of a concept.

4. Direct instruction Activities under Direct instruction

Activities under Concept attainment The teacher will define proper The teacher will give a set of fraction examples. Ex. “A fraction a/b is proper if Ex. “The following are proper lal

Chapter 10; Lesson 2: Guiding Principles/Theories and Teaching Approaches and Methods in the Teaching of Math Grades The Spiral Progression in the Teaching of Math - Basic principles are introduced in the first grade and are rediscovered in succeeding grades in more complex forms - Students continually return to basic ideas as new subjects and concepts are added over the course of a curriculum

7-10 Grades 4-6 Grades 1-3 K– Algebra, Geometry, Statistics And Probability

Three-tiered Principle Theory Definition Jerome Bruner theorized that learning occurs by going through three stages of representation. Each stage is a "way in which information or knowledge are stored and encoded in memory" (Mcleod, 2008). The stages are more-or-less sequential, although they are not necessarily age-related like Piaget-based theories. Going through the stages is essential to truly understanding the concept, as it helps the learner understand why. Bruner's Stages of Representation 1.enactive (action-based) Sometimes called the concrete stage, this first stage involves a tangible hands-on method of learning. Bruner believed that "learning begins with an action - touching, feeling, and manipulating" (Brahier, 2009, p. 52). In

mathematics education, manipulatives are the concrete objects with which the actions are performed. Common examples of manipulatives used in this stage in math education are algebra tiles, paper, coins, etc. - anything tangible. 2. iconic (image-based) Sometimes called the pictoral stage, this second stage involves images or other visuals to represent the concrete situation enacted in the first stage. One way of doing this is to simply draw images of the objects on paper or to picture them in one's head. Other ways could be through the use of shapes, diagrams, and graphs. video: iconic representation. Retreived fromhttp://www.youtube.com/watch? v=gIW0mjMo9IE 3. symbolic (language-based)Sometimes called the abstract stage, the last stage takes the images from the second stage and represents them using words and symbols. The use of words and symbols "allows a student to organize information in the mind by relating concepts together" (Brahier, 2009, p. 53). The words and symbols are abstractions, they do not necessarily have a direct connection to the information. For example, a number is a symbol used to describe how many of something there are, but the number in itself has little meaning without the understanding of it means for there to be that number of something. Other examples would be variables such as x or y, or mathematical symbols such as +, -, /, etc. Finally, language and words are another way to abstractly represent the idea. In the context of math, this could be the use of words such as addition, infinite, the number three, etc

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