TEACHERS >> CREATING A RESULTS-ORIENTED MATH CLASSROOM >>
ORGANIZING LESSONS FOR SUCCESS
©1995-2008 Mahesh C. Sharma All rights reserved.
To have a results-oriented mathematics classroom, we must organize lessons with key mathematics ingredients and components.
A results-oriented lesson includes seven elements:
1. Mental Arithmetic/Mathematics (10 minutes) [Entire Class]
The teacher establishes a strong number sense at each grade level. She achieves these through students counting, estimating, using benchmark numbers ('good' numbers), understanding properties of numbers, using mental arithmetic, and understanding the effect of arithmetic operations on numbers.
þ Counting (forward, backward, skip counting by whole numbers, fractions, and decimals) (choice of number depends on grade level and students)
þ Counting using number grid and number line
þ Oral arithmetic facts and extending facts using larger numbers and associative, distributive and commutative properties
þ Oral mathematics to show patterns, relationships and forming conjectures
þ Problem of the day (individual student)
þ Discussion of the problem (small groups)
2. Linking Concepts and Procedures (10 minutes) [Entire Class]
A mathematics lesson begins with the incorporation of the perspective and knowledge that students bring to the topic. The teacher builds the new concepts and procedures from students’ intuitive and prior knowledge. Each lesson aims at developing multiple, cognitive strategies and focus on depth of the concept, instead of just the quantity of problems. It is better to devote an entire class period to two or three problems from many angles and in-depth, rather than to work on many problems using the same procedure or ‘recipe.’
þ Review of materials and summary of previous day's work (comments and dealing with homework)
þ Multiple assessments
þ Extensions and applications
3. Prerequisite Skills (Embedded in the lesson) [Entire Class and Individuals]
Every lesson includes the development of the following prerequisite and support skills for the concepts being taught:
þ Spatial orientation/space organization
þ Pattern recognition, extensions, applying and creating numerical and geometrical patterns
þ Deductive reasoning
þ Inductive reasoning
·4. New Language, Concept, and/or Procedure (30 minutes) [Entire Class]
The teacher approaches the concept through patterns that show students how to solve problems and see relationships and connections between different concepts and procedures. She uses manipulatives, models and other representations to demonstrate the problem situation, and then links concrete to symbolic and abstract representation. The classroom activities, book exercises and skills pages complement each other and the stated objectives. The teacher accepts multiple correct solutions, and also emphasizes that the answers should aim at the developmental sequence from estimated solutions to exact answers, then to efficient strategies, and, finally, to reach elegant solutions. Enrichment activities are multi-sensory and challenge students.
(a) Concept Building: Levels of Knowing
þ Intuitive (making connections with previous learning; Socratic questioning)
þ Concrete (conceptual and model development; Models?exact, efficient, elegant)
þ Representational (pictorial, graphical, graphing calculator, computer graphics, recording and pre-symbolic)
þ Abstract (recording of the activity, symbolic work on paper, calculator or computer; use of formal language and symbols, tests, quizzes and examinations)
þ Applications (making mathematics relevant?intra-mathematical, interdisciplinary, extracurricular, writing number stories, word problem, projects, modeling, simulations)
þ Communications (demonstration of mastery and competence through written, graphical, compugraphic, concrete, tests, examinations, peer-teaching, designing tests and problems, or oral mathematics.) At the communications level, the teacher should require students to describe and justify their mathematics thinking by using a variety of strategies.
The teacher is aware of and accommodates different mathematics learning
personalities in a classroom, and plans lessons accordingly.
That means the models include:
þ Discrete (quantitative)
þ Continuous (qualitative)
5. Mathematics as a Second Language (Integrated in the lesson)
The teacher bases instruction of the new concept or procedure on situational story problems. From that she brings out the
þ Vocabulary and terminology
þ Structure of mathematical language (syntax)
þ Translation from Math to English (Writing number stories, making conjectures, definitions, articulating patterns)
þ Rranslation from English to Mathematics (Solving word problems)
þ She achieves this through Socratic questioning, scaffolding, and focusing on key questions related to the topic.
6. Review and Consolidation (5 minutes) [Whole Class]
The teacher emphasizes cumulative follow-through in the form of mixed problem-solving, reviews and tests (in multiple forms and modalities) that check on the mastery of specific objectives at different levels of knowing, and makes them accessible to as many students as possible. There is a careful balance between linguistic, conceptual, and procedural knowledge.
The teacher uses ongoing assessments to guide instruction and timeline.
þ Summary and review at the end of the class
þ Summary at the beginning of the next class
þ Repetition, rehearsal, transfer
Each homework assignment has the following components:
þ Cumulative (1/3)
þ Representation of the daily classroom work (1/3)
þ Preview and number stories (1/3)
þ Challenge (1 problem)
Author Contact Information:
Mahesh C. Sharma: Mahesh@mathematicsforall.org
508-877-4089 (H) | 508-494-4608 (C)
Center for Teaching/Learning of Mathematics, Inc.
©1995-2008 Mahesh C. Sharma All rights reserved.