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Math 413 Mathematics Tasks for Cognitive Instruction. October 2008. Connecticut Scope and Sequence Number Sense Operations Estimation Ratio, Proportion and Percent Measurement Spatial Relations and Geometry Probability and Statistics Patterns Algebra and Functions Discrete Mathematics.

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nctm standards compared to connecticut scope and sequence
Connecticut Scope and Sequence

Number Sense

Operations

Estimation

Ratio, Proportion and Percent

Measurement

Spatial Relations and Geometry

Probability and Statistics

Patterns

Algebra and Functions

Discrete Mathematics

NCTM Content Standards

Numbers and Operations

Algebra

Data Analysis and Probability

Geometry

Measurement

NCTM Process Standards

Problem Solving

Reasoning and Proof

Connections

Communication

Representation

NCTM Standards Compared to Connecticut Scope and Sequence
nctm and ct scope and sequence
http://www.nctm.org

http://www.sde.ct.gov/sde/cwp/view.asp?a=2618&q=320872

NCTM and CT Scope and Sequence
the mathematical tasks framework
The Mathematical Tasks Framework

Student

Learning

TASKS

as they

appear in

curricular/

instructional

materials

TASKS

as implemented

by students

TASKS

as set up by

teacher

A representation of how mathematical tasks unfold in the classroom during classroom instruction (Stein & Smith, 1998)

what is a math lesson t h e l e ss o n p l a n
What is a Math Lesson?The Lesson Plan
  • It is a complex function of the work or task that you select
  • How you set it up?
  • How your students will understand what the work you selected demands?
  • What they do?
  • What you make of what they do?
  • What do you do next?
defining levels of cognitive demand of mathematical tasks
Defining Levels of Cognitive Demand of Mathematical Tasks
  • Lower Level Demands
    • Memorization
    • Procedures without connections
  • Higher Level Demands
    • Procedures with Connections
    • Doing Mathematics
verb examples associated with each activity lower level of cognitive demands
Verb Examples Associated with Each Activity Lower Level of Cognitive Demands
  • Knowledge: arrange, define, duplicate, label, list, memorize, name, order, recognize, relate, recall, repeat, reproduce state.
  • Comprehension: classify, describe, discuss, explain, express, identify, indicate, locate, recognize, report, restate, review, select, translate.
defining levels of cognitive demands of mathematical tasks lower level demands
Defining Levels of Cognitive Demands of Mathematical TasksLower Level Demands
  • Memorization:
  • What are the decimal and percent equivalents for the fractions ½ and ¼ ?
defining levels of cognitive demands of mathematical tasks lower level demands10
Defining Levels of Cognitive Demands of Mathematical TasksLower Level Demands
  • Memorization:
  • What are the decimal and percent equivalents for the fractions ½ and ¼ ?
  • Expected Student Response:
  • ½=.5=50%
  • ¼=.25=25%
defining levels of cognitive demands of mathematical tasks lower level demands11
Defining Levels of Cognitive Demands of Mathematical TasksLower Level Demands
  • Procedures without connections:
  • Convert the fraction 3/8 to a decimal and a percent.
  • Expected Student Response:
  • Fraction 3/8
  • Divide 3 by 8 and get a decimal equivalent of .375
  • Move the decimal point two places to the right and get 37.5 %
verb examples associated with each activity higher levels of cognitive demand
Verb Examples Associated with Each Activity Higher levels of cognitive demand
  • Application: apply, choose, demonstrate, dramatize, employ, illustrate, interpret, operate, practice, schedule, sketch, solve, use, write.
  • Analysis: analyze, appraise, calculate, categorize, compare, contrast, criticize, differentiate, discriminate, distinguish, examine, experiment, question, test.
defining levels of cognitive demands of mathematical tasks higher level demands
Defining Levels of Cognitive Demands of Mathematical TasksHigher Level Demands
  • Procedure with connections:
  • Using a 10 by 10 grid, illustrate the decimal and percent equivalents of 3/5.
verb examples associated with each activity highest levels of cognitive demands
Verb Examples Associated with Each ActivityHighest levels of cognitive demands
  • Synthesis: arrange, assemble, collect, compose, construct, create, design, develop, formulate, manage, organize, plan, prepare, propose, set up, write.
  • Evaluation: appraise, argue, assess, attach, choose, compare, defend estimate, judge, predict, rate, core, select, support, value, evaluate
defining levels of cognitive demands of mathematical tasks higher level demands15
Defining Levels of Cognitive Demands of Mathematical TasksHigher Level Demands
  • Doing Mathematics:
  • Shade 6 small squares in a 4 X 10 rectangle. Using the rectangle, explain how to determine each of the following:
  • A) the percent of area that is shaded
  • B) the decimal part of the area that is shaded
  • C) the fractional part of the area that is shaded
lesson planning
Lesson Planning
  • An effective lesson plan begins with a relevant clearly written objective.
lesson objective definition and purpose
Lesson ObjectiveDefinition and Purpose
  • An objective is a description of a learning outcome.
  • Objectives describe where we want students to go – not how they will get there.
  • Well written objectives clarify what teachers want their students to learn, help provide lesson focus and direction, and help guide the selection of appropriate practice.
  • In addition, teachers can assess their students learning and their own teaching to determine if the lesson objective has been met
example of a state standard content standard 1 grade 4
Example of a State StandardContent Standard#1 Grade 4
  • Use models, benchmarks and equivalent forms to judge the size of fractions (in relation to ½,1/4, ¾ and the whole and decimals in situations relevant to students’)
number and operations standard for grades 3 5 expectations example of the nctm standard
Number and Operations Standard for Grades 3-5 Expectations Example of the NCTM Standard
  • In grades 3-5 all students should-Use models, benchmarks, and equivalent forms to judge the size of fractions
  • Recognize and generate equivalent forms of commonly used fractions, decimals and percents
  • http://www.nctm.org
from general to specific going from state standards to objectives
From General to Specific: Going from State Standards to Objectives
  • While state and national standards provide general content ideas, teachers are responsible for writing their own objectives for their lessons, activities and units.
  • A teacher’s job is to translate the standards into useful objectives that are used to guide instruction.
  • The learning outcomes included in the objectives will then be linked to the state standards.
how standards goals and objectives differ
How standards, goals, and objectives differ…
  • Specific –Objectives include specific learning outcomes where standards include general outcome statements.
  • Goals may be general, for example, understand the concept of fractions.
  • Long-Term or Short Term –Objectives are considered short term, they describe the learning outcome typically in days, or weeks.
  • Goals and standards describe learning outcomes that may be in weeks, months or years.
how standards goals and objectives differ22
How standards, goals, and objectives differ…
  • Uses – Objectives are used in lesson and activity plans and IEPs.
  • Measurable annual goals are included in IEPs.
  • Goals may also be found in units of instruction. For example, a goal may be to understand how to add fractions.
  • A specific objective may be to be able to add fractions will common denominators.
examples of goals and objectives related to state standards
Examples of Goals and Objectives Related to State Standards
  • Students will write answers to 20 subtraction problems (two-digit numbers from three-digit numbers with re-grouping) on a worksheet, with two errors.
the four components of an objective
The Four Components of an Objective
  • Content
  • Behavior
  • Condition
  • Criterion
  • Concepts taken from Daily Planning for Today’s Classroom by Kay M. Price and Karna L. Nelson
content
Content
  • Students will write answers to 20 subtraction problems (two-digit numbers from three-digit numbers with re-grouping) on a worksheet, with two errors.
  • Content- In the example given the contentissubtraction problems (two-digit numbers from three-digit numbers with re-grouping)
behavior
Behavior
  • Students will write answers to 20 subtraction problems (two-digit numbers from three-digit numbers with re-grouping) on a worksheet, with two errors.
  • Behavior- the behavior tells what the students will do to show that they have learned.
  • It is a verb that describes an observable action. In this example the behavior is “write”. The student will demonstrate knowledge of subtraction by writing the answers to the 20 problems. (See Bloom)
condition
Condition
  • Condition-It is important to describe the conditions or circumstances under which the student will perform the behavior.
  • In the example objective, the condition is “on a worksheet” not in a real world context.
criterion
Criterion
  • Students will write answers to 20 subtraction problems (two-digit numbers from three-digit numbers with re-grouping) on a worksheet, with two errors.
  • Criterion-The criterion is the level of acceptance performance, the standard of mastery of proficiency level expected.
  • In the objective above, the criterion is with two errors.
examples and nonexamples of content
Examples and Nonexamples of Content
  • Add unlike fractions with common factors between denominators
  • ________________________________
  • Add fractions on page 42, 1 to 7
examples and nonexamples of behavior
Examples and Nonexamples of Behavior
  • Diagram, operate, order, compare/contrast
  • _________________________________
  • Know, understand, memorize, learn
examples and nonexamples of conditions
Examples and Nonexamples of Conditions
  • Given ten problems and a calculator
  • __________________________________
  • Given a blank piece of paper, when asked by the teacher (obvious)
examples and nonexamples of criterion
Examples and Nonexamples of Criterion
  • With no errors
  • With 80 percent accuracy
  • Within 10 minutes
  • To the nearest tenth
  • __________________________
  • As judged by the teacher
  • To the teacher’s satisfaction
a final thought
A Final Thought
  • It is very important to begin your lesson or activity with a clear idea of what you want your students to learn.
  • Writing a specific objective with the four components will cause you to think this through.
  • When teachers experience frustration with a particular lesson, they often have not stated a measurable objective.
  • If you clearly state the objective, you will know if your activity or lesson and your intended learning outcome match. You will be able to tell if your teaching was effective and whether your students learned.