Competency 14: Mathematics Instruction Flashcards
T o F: Research supports conceptually-base teaching, or teaching concepts rather than just memorizing facts.
True. Students should first build an understanding of what the concept of multiplication actually is before learning the multiplication table.
Individual Education Plans (IEPs)
legal requirements for special education students.
Mathematics should be taught
conceptually
Manipulatives
Any objects that can be touched or moved to assist understanding. Can take the form of counters such as small toys or buttons used to help a student add. EX) 100 noodles glued onto construction paper to show place value, or a circle folded & cut into eighths to assist with understanding fractions.
David Ausabel
cognitive psychologist who is credited with the theory of Advanced Organizers. He believed that in order for new information to be learned, it must be put into a context or linked to knowledge that already exists. Meaningful learning is considered to be new knowledge that relates to what is already known. Advocated ‘anchoring idea’
New information should be organized to answer 3 simple questions:
1) What will the person learn?
2) What does the person want to learn?
3) What did the person learn?
Benjamin Bloom
Educational psychologist who made contributions to the classification of educational objectives and to the theory of mastery-learning. Outlined a classification of learning objectives (Blooms Taxonomy)
William Brownell
Educational psychologist. Mathematics
Robert Gagne
educational psychologist best known for his "Conditions of Learning." Pioneered science of instruction. Assumption that different types of learning exist, and that different instuctional conditions are most likely to bring about these different types of learning. There are several types & levels of learning & each of these levels requires instruction that is tailored to meet the needs of the pupil. Focus of theory is on the retention and honing of intellectial skills. 1) Intellectual skills 2) Cognitive strategies 3) Verbal information 4) Motor skills 5) Attitudes 8 ways to learn 1) Signal Learning 2) Stimulus-Response Learning 3) Chaining 4)Verbal Association 5) Discrimination Learning 6) Concept Learning 7) Rule Learning 8) Problem Solving
Jean Piaget
Educational/clinical psychologist. who placed great importance on the education of children. Famous for theory of cognitive development. Ones childhood lays a vital & active role in a person’s development. Believed in child-centered classrooms and open education.
Lee Vygotsky
Developmental psychologist. Proposed theory of the develpment of higher cognitive functions in children that saw reasoning as emerging through practical activity in a social environment. Concept of the zone of proximal development. The aquisition of new knowlege is dependent on previous learning and the availability of instruction.
Bloom’s Taxonomy
Set of 3 hierarchical models used to classify educational learning objectives into levels of complexity and mastery. The 3 lists cover the learning objectives in cognitive, affective and sensory domains.
Piaget’s theory of cognitive development
Developmental stage theory about the nature & development of human intelligence. 1) Sensorimotor stage (0-2
2) Preoperational stage (2-7)
3) Concrete operational stage (7-11)
4) Formal operational stage (11-16)
Sensorimotor Stage
Stage 1 of Piaget’s theory of cognitive development. Occurs from birth to 2 (acquisition of language) Construct knowledge by coordinating experiences with physical interactions with objects (grasping, sucking, stepping). Progress from reflexive, instinctual action at birth to the beginning of symbolic thought toward the end of the stage. Children learn they are separated from the environment. Development of object permanence, a child’s understanding that objects continue to exist even though he cannot see or hear it, is one of the most important accomplishments. (Peek-a-boo) By the end of this stage, children develop a permanent sense of self & object.
Pre-operational Stage
Stage 2 of Piaget’s theory of cognitive development. Occurs from 2-7. Starts when the child begins to learn to speak. Children do not yet understand concrete logic & cannot mentally manipulate information. Increase in playing & pretending. Has trouble seeing things from different points of view. Symbolic play & manipulating symbols. Ex) checkers being snacks, pieces of paper being plates, and a box being a table. Child able to form stable concepts & magical beliefs. Not able to perform operations. Egocentric thinking. Able to understand/represent/remember/ picture objects in their mind without having it in front of them. Children propose questions of why and how come? Children want to knowledge of knowing everything.
Concrete operational stage
Stage 3 of Piaget’s theory of cognitive development. Occurs from 7-11. Characterized by appropriate use of logic. A child’s thought process becomes more mature & adult like. Start solving problems in a more logical fashion. Abstract, hypothetical thinking not yet developed. Children can only solve problems that apply to concrete events or objects. Children undergo a transition where the child learns rules such as conservation. Children able to incorporate inductive reasoning: involves drawing ingerencs from observations in order to make a generalization. Children struggle with deductive reasoning, which involves using a generalized principle in order to try to predict the outcome of an event. Children commonly experienc difficulties with figuring out logic in their heads. Logic & elimination of egocentrism.
Formal Operational stage
Stage 4 of Piaget’s theory of cognitive development. Occurs from 11-above. Intelligence demonstrated through logical use of symbols related to abstract concepts. Capable of hypothetical & deductive reasoning. Able to think about abstract concepts. What-if situations. Abstract thought, metagognition, problem-solving.
Learning Modalities
Visual, Auditory, Kinesthetic
Association
Words or symbols Ex) even young children associate word triangle with the shape.
Concept
Relational or concrete attributes. Ex) Similar figures have relational attributes. The corresponding angles are equal and the ratios or corresponding sides are equal.
Principle
Generalizations, developed rules. Ex) The area of a trapezoid is developed from the concept of a trapezoid and the area of triangles. rectangles, and/or parallelograms.
Problem Solving
putting together concepts & principles to solve a problem new to the learner.
Instructional Moves
Linked between and among the following using modeling, describing and recording by both teacher & student:
Concrete- manipulatives, models, hands-on.
Pictorial- pics, diagrams, graphs, tech.
Abstract- symbols, words.
General problem-solving method
1) Understand
2) Plan
3) Solve
4) Check
Inductive (informal) reasoning
The reasoning goes from the specific to the general. It uses observations and patterns to infer a generalization.
Deductive (formal) reasoning
The reasoning process reaches conclusions based on accepted truths and logical reasoning. The reasoning goes from the general to the specific, and uses a general rule or statement to draw a reasonable conclusion.
Formal Asessment
An end-product testing. Ex) a unit test, a final exam.
The familiar paper-and-pencil tests such as teacher-made tests/quizzes, but can also be diagnostic screenings, standardized tests, spelling tests.
Informal Asessment
The teacher observes, listens, and questions students in order to gather information regarding student learning. EX) when conducting guided practice, the teacher observes that 3 students were counting using their fingers. ex) teacher observations, graded daily work, student answers to oral questions.
Diagnostic Assessment
Used to predetermine student’s knowledge. Ex) pre-test before instruction on a unit has begun.
Formative Assessment
Used to determine students’ ongoing learning & retention, whether or not additional instruction is needed. Ex) Giving a quiz within a unit.
Summative Assessment
Used to determine mastery of material. EX) A semester exam or a project, incorporating all objectives from a unit.
Standardized Assessment
Written to general content, and the performance on the test is based on a comparison to other similar students who took the test. EX) SAT, ACT, GRE.
Criterion Referenced Assessment
Written to measure specific content and the criteria for passing the test is pre-specified. EX) Well-designed teacher unit test, STAAR, TExES.
